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                    Mathematica-Compatible Notebook

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Cell[BoxData[
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Numbers other than integers can be represented in various bases. \
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Cell["Following this command, the earlier context is restored. ", \
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Cell[CellGroupData[{

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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
This introduces the objects intended to be \
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\>", "ExampleText",
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"BeginPackage"],

Cell[BoxData[
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}, Open  ]],

Cell["Here is the definition of the function. ", "ExampleText",
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Cell[BoxData[
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    Collatz[n_Integer] := 
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Here is a table of the first nine rows of Pascal's triangle. \
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["ButtonBox", "ObjectName",
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  ButtonBox["ButtonNote",
    ButtonStyle->"RefGuideLink"],
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  ButtonBox["ButtonStyle",
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Cell[BoxData[
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\\\"MR\\\"]\\) \"\>", "\<\"the front end \"\>"},
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  ButtonBox["ButtonStyle",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["SelectionEvaluate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ButtonNotebook",
    ButtonStyle->"RefGuideLink"],
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Cell["ButtonExpandable", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  ButtonBox["ButtonMinHeight",
    ButtonStyle->"RefGuideLink"],
  ", ",
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    ButtonStyle->"RefGuideLink"],
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}], "Notes",
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Cell["ButtonFrame", "ObjectName",
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Cell[TextData[{
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}], "Usage",
  CellTags->{"S0.1056", "ButtonFrame"}],

Cell["\<\
\[FilledSmallSquare] Typical settings supported include: \
\>", \
"Notes",
  CellTags->{"S0.1056", "ButtonFrame"}],

Cell[BoxData[
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\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"a button in a palette \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"DialogBox\\\\\\\
\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"a button in a dialog box \
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Cell[TextData[{
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  " may look slightly different on different computer systems. \n\
\[FilledSmallSquare] See ",
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  ButtonBox["Section 2.8.11",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["ButtonStyle",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["WindowFrame",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1056", "ButtonFrame"}],

Cell["ButtonFunction", "ObjectName",
  CellTags->{"S0.1056", "ButtonFunction"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["ButtonFunction", "MR"],
  " is an option for ",
  StyleBox["ButtonBox", "MR"],
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clicked on. "
}], "Usage",
  CellTags->{"S0.1056", "ButtonFunction"}],

Cell[TextData[{
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\[FilledSmallSquare] With ",
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  Cell[BoxData[
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  " for all ",
  Cell[BoxData[
      \(TraditionalForm\`k\)], "InlineFormula"],
  " relatively prime to ",
  Cell[BoxData[
      \(TraditionalForm\`n\)], "InlineFormula"],
  ". "
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  CellTags->{"S0.1059", "CarmichaelLambda"}],

Cell[TextData[{
  "\[FilledSmallSquare] Integer mathematical function (see Section ",
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    ButtonData:>{"A.3.10", "10.1"},
    ButtonStyle->"MainBookLink"],
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  StyleBox["CarmichaelLambda", "MR"],
  " returns unevaluated if there is no integer ",
  Cell[BoxData[
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  " satisfying the necessary conditions. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.2.4",
    ButtonData:>{"3.2.4", "4.30"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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  ButtonBox["EulerPhi",
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  ", ",
  ButtonBox["RealDigits",
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}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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  Cell[BoxData[
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  Cell[BoxData[
      \(TraditionalForm\`6\)], "InlineFormula"],
  ". The following examples show what this means. "
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Cell[BoxData[
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  CellLabel->"In[1]:= ",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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  Cell[BoxData[
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  Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[3]= ",
  CellTags->"CarmichaelLambda"]
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Cell[TextData[{
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  Cell[BoxData[
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  " modulo ",
  Cell[BoxData[
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  ". "
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  CellTags->"CarmichaelLambda"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"CarmichaelLambda"],

Cell[BoxData[
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  CellLabel->"Out[4]= ",
  CellTags->"CarmichaelLambda"]
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellTags->"CarmichaelLambda"]
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Cell[BoxData[
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  CellLabel->"In[6]:= ",
  CellTags->"CarmichaelLambda"],

Cell["Cases", "ObjectName",
  CellTags->{"S0.1060", "Cases"}],

Cell[TextData[{
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
  " ",
  StyleBox["pattern", "TI"],
  StyleBox["]", "MR"],
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  Cell[BoxData[
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  StyleBox["Cases[", "MR"],
  Cell[BoxData[
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  Cell[BoxData[
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
  " ",
  StyleBox["pattern", "TI"],
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  StyleBox["->", "MR"],
  " ",
  StyleBox["rhs", "TI"],
  StyleBox["]", "MR"],
  " gives a list of the values of ",
  StyleBox["rhs", "TI"],
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  Cell[BoxData[
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  StyleBox["Cases[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["pattern", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["levspec", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["expr", "TI"],
  " on levels specified by ",
  StyleBox["levspec", "TI"],
  " which match the pattern. \n\[FilledSmallSquare] ",
  StyleBox["Cases[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["pattern", "TI"],
  " ",
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  StyleBox["rhs", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["levspec", "TI"],
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  StyleBox["rhs", "TI"],
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  StyleBox["2, x, 4", "MR"],
  Cell[BoxData[
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  StyleBox["List", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Cases[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["pattern", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["levspec", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
  " gives the first ",
  StyleBox["n", "TI"],
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  StyleBox["expr", "TI"],
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  ButtonBox["Section 2.3.2",
    ButtonData:>{"2.3.2", "2.2"},
    ButtonStyle->"MainBookLink"],
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  ButtonBox["Position",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ReplaceList",
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  ButtonBox["Collect",
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  ButtonBox["Statistics`DataManipulation`",
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    ButtonStyle->"AddOnsLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1060", "Cases"}]
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"Cases"],

Cell["This command returns the summands that are powers. ", "ExampleText",
  CellTags->"Cases"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Cases[
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  CellLabel->"In[1]:= ",
  CellTags->"Cases"],

Cell[BoxData[
    \({\[ExponentialE]\^\(-x\^2\), \[ExponentialE]\^\(y\^2\), u\^2, 
      w\^\(5\^x\), 1\/z\^2}\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"Cases"]
}, Open  ]],

Cell[TextData[{
  "This returns the expressions that occur in the sum at any level and are \
powers. Note how ",
  Cell[BoxData[
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  " matches ",
  StyleBox["x_^y_", "MR"],
  " in one occurrence and ",
  StyleBox["y_", "MR"],
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  CellTags->"Cases"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Cases"],

Cell[BoxData[
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  CellLabel->"Out[2]= ",
  CellTags->"Cases"]
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Cell[TextData[{
  StyleBox["Cases", "MR"],
  " can also perform an arbitrary replacement in the terms that match the \
given pattern. Here we pick all multiples of three, replacing them by their \
squares. "
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  CellTags->"Cases"],

Cell[CellGroupData[{

Cell[BoxData[
    \(L = Array[Random[Integer, 10] &, 20]; Print[L]; 
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  CellLabel->"In[3]:= ",
  CellTags->"Cases"],

Cell[BoxData[
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  CellTags->"Cases"],

Cell[BoxData[
    \({81, 36, 0, 81, 36, 9}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"Cases"]
}, Open  ]],

Cell[TextData[{
  "This substitution can be useful, for example, in controlling recursion. \
Here is a set of rules that define a finite\[Hyphen]state automaton with six \
states and a four\[Hyphen]input alphabet. (The first argument of ",
  StyleBox["fsa", "MR"],
  " is the state, and the second is the input; the return value is the state \
to which the automaton moves. The 0 state indicates failure, and 1 is the \
initial state.) "
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  CellTags->"Cases"],

Cell[BoxData[
    \(fsa[1, 1] = 2; fsa[1, 2] = 3; fsa[1, 3] = 4; fsa[1, 4] = 5; 
    fsa[2, 1] = 2; fsa[2, 2] = 3; fsa[2, 3] = 0; fsa[2, 4] = 5; 
    fsa[3, 1] = 2; fsa[3, 2] = 3; fsa[3, 3] = 4; fsa[3, 4] = 0; 
    fsa[4, 1] = 0; fsa[4, 2] = 3; fsa[4, 3] = 4; fsa[4, 4] = 5; 
    fsa[5, 1] = 2; fsa[5, 2] = 0; fsa[5, 3] = 4; fsa[5, 4] = 5; 
    fsa[6, 1] = 0; fsa[6, 2] = 0; fsa[6, 3] = 0; fsa[6, 4] = 0; start = 1; 
    failure = 0; \)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"Cases"],

Cell[TextData[{
  "Suppose we want to list all strings of length at most ",
  StyleBox["maxlen", "MR"],
  " that are accepted by the automaton, that is, that do not take the \
automaton to the failure state. The function ",
  StyleBox["Recurse", "MR"],
  " does the job using ",
  StyleBox["Cases", "MR"],
  " (one might instead use ",
  StyleBox["Do", "MR"],
  " or ",
  StyleBox["Table", "MR"],
  "). "
}], "ExampleText",
  CellTags->"Cases"],

Cell[BoxData[
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            Recurse[history <> ToString[inputChar], fsa[state, inputChar]]], 
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  CellLabel->"In[5]:= ",
  CellTags->"Cases"],

Cell[TextData[{
  "The function ",
  StyleBox["Useful", "MR"],
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  CellTags->"Cases"],

Cell[BoxData[
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  CellLabel->"In[6]:= ",
  CellTags->"Cases"],

Cell[TextData[{
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  StyleBox["maxlen=4", "MR"],
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  CellTags->"Cases"],

Cell[CellGroupData[{

Cell[BoxData[
    \(maxlen = 4; Recurse["\<\>", start]\)], "Input",
  CellLabel->"In[7]:= ",
  CellTags->"Cases"],

Cell[BoxData[
    \({\*"\<\"\"\>", {"1", {"11", {"111", {"1111"}, {"1112"}, {"1114"}}, \
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{"1144"}}}, {"12", {"121", {"1211"}, {"1212"}, {"1214"}}, {"122", {"1221"}, \
{"1222"}, {"1223"}}, {"123", {"1232"}, {"1233"}, {"1234"}}}, {"14", {"141", \
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{"1441"}, {"1443"}, {"1444"}}}}, {"2", {"21", {"211", {"2111"}, {"2112"}, \
{"2114"}}, {"212", {"2121"}, {"2122"}, {"2123"}}, {"214", {"2141"}, {"2143"}, \
{"2144"}}}, {"22", {"221", {"2211"}, {"2212"}, {"2214"}}, {"222", {"2221"}, \
{"2222"}, {"2223"}}, {"223", {"2232"}, {"2233"}, {"2234"}}}, {"23", {"232", \
{"2321"}, {"2322"}, {"2323"}}, {"233", {"2332"}, {"2333"}, {"2334"}}, {"234", \
{"2341"}, {"2343"}, {"2344"}}}}, {"3", {"32", {"321", {"3211"}, {"3212"}, \
{"3214"}}, {"322", {"3221"}, {"3222"}, {"3223"}}, {"323", {"3232"}, {"3233"}, \
{"3234"}}}, {"33", {"332", {"3321"}, {"3322"}, {"3323"}}, {"333", {"3332"}, \
{"3333"}, {"3334"}}, {"334", {"3341"}, {"3343"}, {"3344"}}}, {"34", {"341", \
{"3411"}, {"3412"}, {"3414"}}, {"343", {"3432"}, {"3433"}, {"3434"}}, {"344", \
{"3441"}, {"3443"}, {"3444"}}}}, {"4", {"41", {"411", {"4111"}, {"4112"}, \
{"4114"}}, {"412", {"4121"}, {"4122"}, {"4123"}}, {"414", {"4141"}, {"4143"}, \
{"4144"}}}, {"43", {"432", {"4321"}, {"4322"}, {"4323"}}, {"433", {"4332"}, \
{"4333"}, {"4334"}}, {"434", {"4341"}, {"4343"}, {"4344"}}}, {"44", {"441", \
{"4411"}, {"4412"}, {"4414"}}, {"443", {"4432"}, {"4433"}, {"4434"}}, {"444", \
{"4441"}, {"4443"}, {"4444"}}}}}\)], "Output",
  CellLabel->"Out[7]= ",
  CellTags->"Cases"]
}, Open  ]],

Cell[BoxData[
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  CellLabel->"In[8]:= ",
  CellTags->"Cases"],

Cell[TextData[{
  " (See also the Further Examples for the ",
  ButtonBox["Heads",
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Cell["Catalan", "ObjectName",
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Cell[TextData[{
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  Cell[BoxData[
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  CellTags->{"S0.1061", "Catalan"}],

Cell[TextData[{
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  ButtonBox["A.3.11",
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          UnderoverscriptBox["\[Sum]", \(k = 0\), "\[Infinity]",
            
            LimitsPositioning->
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  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.2.8",
    ButtonData:>{"3.2.8", "8.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] Implementation Notes: see section ",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell["Here is a numerical approximation to the Catalan constant. ", \
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Catalan"],

Cell[BoxData[
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  CellLabel->"Out[1]= ",
  CellTags->"Catalan"]
}, Open  ]],

Cell["Catch", "ObjectName",
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Cell[TextData[{
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  StyleBox["]", "MR"],
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  StyleBox["Throw", "MR"],
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  StyleBox["expr", "TI"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Catch[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["form", "TI"],
  StyleBox["]", "MR"],
  " returns ",
  StyleBox["value", "TI"],
  " from the first ",
  StyleBox["Throw[", "MR"],
  StyleBox["value", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["tag", "TI"],
  StyleBox["]", "MR"],
  " for which ",
  StyleBox["form", "TI"],
  " matches ",
  StyleBox["tag", "TI"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Catch[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["form", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["f", "TI"],
  StyleBox["]", "MR"],
  " returns ",
  StyleBox["f", "TI"],
  StyleBox["[", "MR"],
  StyleBox["value", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["tag", "TI"],
  StyleBox["]", "MR"],
  ". "
}], "Usage",
  CellTags->{"S0.1061", "Catch"}],

Cell[TextData[{
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  StyleBox["Catch[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["]", "MR"],
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  StyleBox["expr", "TI"],
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  StyleBox["Throw", "MR"],
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  StyleBox["form", "TI"],
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  StyleBox["tag", "TI"],
  " in ",
  StyleBox["Throw[", "MR"],
  StyleBox["value", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["tag", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["form", "TI"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.5.9",
    ButtonData:>{"2.5.9", "9.18"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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  ButtonBox["CheckAbort",
    ButtonStyle->"RefGuideLink"],
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  CellTags->{"S0.1061", "Catch"}]
}, Closed]],

Cell[CellGroupData[{

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Cell[TextData[{
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}], "ExampleText",
  CellTags->"Catch"],

Cell[BoxData[
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  CellTags->"Catch"],

Cell[TextData[{
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  StyleBox["overflow", "MR"],
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  StyleBox["Catch", "MR"],
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}], "ExampleText",
  CellTags->"Catch"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[2]:= ",
  CellTags->"Catch"],

Cell[BoxData[
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  CellLabel->"Out[2]= ",
  CellTags->"Catch"]
}, Open  ]],

Cell[TextData[{
  "In this case, ",
  StyleBox["overflow", "MR"],
  " is not generated, and the ",
  StyleBox["Catch", "MR"],
  " has no effect. "
}], "ExampleText",
  CellTags->"Catch"],

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                "InlineFormula"], "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"bottom\\\
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  Cell[BoxData[
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",
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  Cell[BoxData[
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Cell["CellGrouping", "ObjectName",
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Cell[TextData[{
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  ButtonBox["Section 2.10.13",
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Cell["CellLabel", "ObjectName",
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Cell[TextData[{
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  StyleBox["CellLabelAutoDelete->True", "MR"],
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  StyleBox["The Mathematica Book: ", "TI"],
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Cell["CellLabelAutoDelete", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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Cell["CellMargins", "ObjectName",
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Cell[TextData[{
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                "InlineFormula"], "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"bottom\\\
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Cell[TextData[{
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\[FilledSmallSquare] See ",
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Cell["CellOpen", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["The Mathematica Book: ", "TI"],
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  ButtonBox["CellGroupData",
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Cell["CellPrint", "ObjectName",
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Cell[TextData[{
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\[FilledSmallSquare] ",
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  Cell[BoxData[
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"cell\\\\\\\"\\\", \
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  CellTags->{"S0.1061", "CellPrint"}],

Cell[TextData[{
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  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
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  ButtonBox["Print",
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  ButtonBox["NotebookWrite",
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  ButtonBox["NotebookPrint",
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Cell["CellTags", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  ButtonBox["Section 2.10.3",
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  ButtonBox["ConversionRules",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
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Cell["CForm", "ObjectName",
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Cell[TextData[{
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}], "Usage",
  CellTags->{"S0.1062", "CForm"}],

Cell[TextData[{
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structures are translated. \n\[FilledSmallSquare] No declarations are \
generated. \n\[FilledSmallSquare] ",
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affects printing, but not evaluation. \n\[FilledSmallSquare] See ",
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  ButtonBox["Compile",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1062", "CForm"}]
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"CForm"],

Cell["Power(x,2) + 2*x*Sqrt(y) + y", "Output",
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  CellTags->"CForm"]
}, Open  ]],

Cell["Character", "ObjectName",
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Cell[TextData[{
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  StyleBox["Read", "MR"],
  ". "
}], "Usage",
  CellTags->{"S0.1063", "Character"}],

Cell[TextData[{
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    ButtonStyle->"MainBookLink"],
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] Related package: ",
  ButtonBox["Utilities`BinaryFiles`",
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    ButtonStyle->"AddOnsLink"],
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}], "Notes",
  CellTags->{"S0.1063", "Character"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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}], "ExampleText",
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Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"Character"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(!! strings\)\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"Character"],

Cell["\"Here is a text string\"", "Print",
  CellTags->"Character"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[3]:= ",
  CellTags->"Character"],

Cell[BoxData[
    \({"\"", "H", "e", "r", "e", " ", "i", "s", " ", "a", " ", "t", "e", "x", 
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  CellTags->"Character"]
}, Open  ]],

Cell["CharacterEncoding", "ObjectName",
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Cell[TextData[{
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character encoding should be used. "
}], "Usage",
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Cell[TextData[{
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  ". \n\[FilledSmallSquare] See ",
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  ButtonBox["Section 2.7.8",
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    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.11.3",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["ToCharacterCode",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FromCharacterCode",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["StringReplace",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$SystemCharacterEncoding",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ShowSpecialCharacters",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1063", "CharacterEncoding"}],

Cell["CharacterRange", "ObjectName",
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Cell[TextData[{
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  StyleBox["\",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"c\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox["\"]", "MR"],
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  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"c\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
  StyleBox["\"", "MR"],
  " to ",
  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"c\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox["\"", "MR"],
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}], "Usage",
  CellTags->{"S0.1063", "CharacterRange"}],

Cell[TextData[{
  "\[FilledSmallSquare] Example: ",
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  Cell[BoxData[
      InterpretationBox[
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          NumberMarks->True],
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  CellLabel->"In[2]:= ",
  CellTags->"Coefficient"],

Cell[BoxData[
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  CellTags->"Coefficient"]
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Cell[TextData[{
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  Cell[BoxData[
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  ". "
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Coefficient"],

Cell[BoxData[
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  CellTags->"Coefficient"]
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Cell[TextData[{
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  StyleBox["x", "MR"],
  " ",
  StyleBox["z", "MR"],
  ", ",
  StyleBox["y", "MR"],
  " and ",
  StyleBox["z", "MR"],
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[4]:= ",
  CellTags->"Coefficient"],

Cell[BoxData[
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  CellLabel->"Out[4]= ",
  CellTags->"Coefficient"]
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Cell["You can use variables that are powers. ", "ExampleText",
  CellTags->"Coefficient"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Coefficient"],

Cell[BoxData[
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  CellLabel->"Out[5]= ",
  CellTags->"Coefficient"]
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Cell["Such a power does not necessarily have to be an integer. ", \
"ExampleText",
  CellTags->"Coefficient"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Coefficient"],

Cell[BoxData[
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  CellLabel->"Out[6]= ",
  CellTags->"Coefficient"]
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Cell["\<\
Coefficient is quite fast for polynomial input. (The time will vary \
for different machines.) \
\>", "ExampleText",
  CellTags->"Coefficient"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Coefficient"],

Cell[BoxData[
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        24\ \((71394081804853879892294324526776320 + 
              1981583836043018240\ y)\) + 
        8\ \((4297574794604266649036299324298884544958662761271787520 + 
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              354455808672194887680\ y\^2)\)}\)], "Output",
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  CellTags->"Coefficient"]
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Cell["Even this harder example comes back in reasonable time. ", "ExampleText",
  CellTags->"Coefficient"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["Coefficient can handle some algebraic dependencies. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["CoefficientList", "ObjectName",
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Cell[TextData[{
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  StyleBox["CoefficientList[", "MR"],
  StyleBox["poly", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["var", "TI"],
  StyleBox["]", "MR"],
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  " in ",
  StyleBox["poly", "TI"],
  ", starting with power ",
  StyleBox["0", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["CoefficientList[", "MR"],
  StyleBox["poly", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"var\\\\\\\"\\\", \
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  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]", "MR"],
  " gives an array of coefficients of the ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"var\\\\\\\"\\\", \
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Cell[TextData[{
  "\[FilledSmallSquare] Example: ",
  StyleBox["CoefficientList[x^2 + 2 x y - y, ", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["x, y", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]\[LongRightArrow]", "MR"],
  Cell[BoxData[
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  ". \n\[FilledSmallSquare] The dimensions of the array returned by ",
  StyleBox["CoefficientList", "MR"],
  " are determined by the values of the ",
  StyleBox["Exponent[", "MR"],
  StyleBox["poly", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"var\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "i"], TraditionalForm]], "InlineFormula"],
  StyleBox["]", "MR"],
  ". \n\[FilledSmallSquare] Terms that do not contain positive integer powers \
of a particular variable are included in the first element of the list for \
that variable. \n\[FilledSmallSquare] ",
  StyleBox["CoefficientList", "MR"],
  " always returns a full rectangular array. Combinations of powers that do \
not appear in ",
  StyleBox["poly", "TI"],
  " give zeros in the array. \n\[FilledSmallSquare] ",
  StyleBox["CoefficientList", "MR"],
  " works whether or not ",
  StyleBox["poly", "TI"],
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  ButtonBox["Section 3.3.2",
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  ", ",
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  ", ",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"CoefficientList"],

Cell[TextData[{
  "This gives the coefficients in the polynomial first in the variable ",
  StyleBox["x", "MR"],
  ". "
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  CellTags->"CoefficientList"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"CoefficientList"],

Cell[BoxData[
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  CellLabel->"Out[1]= ",
  CellTags->"CoefficientList"]
}, Open  ]],

Cell[TextData[{
  "On the other hand this gives the coefficients in the polynomial first in \
the variable ",
  StyleBox["y", "MR"],
  ". "
}], "ExampleText",
  CellTags->"CoefficientList"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CoefficientList[x\^4 + 2  x\^3\ y + 5  x\ y\^2 + 13  y\^4, 
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  CellLabel->"In[2]:= ",
  CellTags->"CoefficientList"],

Cell[BoxData[
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  CellLabel->"Out[2]= ",
  CellTags->"CoefficientList"]
}, Open  ]],

Cell["\<\
We can also get the list of coefficients over the integers modulo 3 \
and 4. \
\>", "ExampleText",
  CellTags->"CoefficientList"],

Cell[CellGroupData[{

Cell[BoxData[
    \({CoefficientList[x\^4 + 2  x\^3\ y + 5  x\ y\^2 + 13  y\^4, x, \ 
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      CoefficientList[x\^4 + 2  x\^3\ y + 5  x\ y\^2 + 13  y\^4, x, \ 
        Modulus \[Rule] 4]}\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"CoefficientList"],

Cell[BoxData[
    \({{y\^4, 2\ y\^2, 0, 2\ y, 1}, {y\^4, y\^2, 0, 2\ y, 1}}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"CoefficientList"]
}, Open  ]],

Cell["\<\
CoefficientList can handle powers of variables in the second \
argument as long as the first argument is a polynomial in those powers. \
\
\>", "ExampleText",
  CellTags->"CoefficientList"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CoefficientList[x\^\(1/2\) + 2  x\^2\  + \ 3  x\^4, \ 
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  CellLabel->"In[4]:= ",
  CellTags->"CoefficientList"],

Cell[BoxData[
    \({0, 1, 0, 0, 2, 0, 0, 0, 3}\)], "Output",
  CellLabel->"Out[4]= ",
  CellTags->"CoefficientList"]
}, Open  ]],

Cell["\<\
The coefficients of a polynomial in two variables are given as an \
array. \
\>", "ExampleText",
  CellTags->"CoefficientList"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[5]:= ",
  CellTags->"CoefficientList"],

Cell[BoxData[
    \({{f, 0, 0, 0}, {0, 0, e, d}, {0, 0, c, 0}, {0, b, 0, 0}, {a, 0, 0, 
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  CellLabel->"Out[5]= ",
  CellTags->"CoefficientList"]
}, Open  ]],

Cell["\<\
Here are the coefficients of some polynomials in three variables. \
\
\>", "ExampleText",
  CellTags->"CoefficientList"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CoefficientList[\(a\_\(2, 2, 0\)\) \(x\^2\) y\^2 + \(a\_\(1, 2, 1\)\) 
            x\ \(y\^2\) z + \(a\_\(2, 0, 1\)\) \(x\^2\) 
            z + \(a\_\(3, 0, 1\)\) \(x\^3\) z + \(a\_\(0, 0, 1\)\) 
            z + \(a\_\(1, 1, 1\)\) x\ y\ z + \(a\_\(1, 0, 1\)\) 
            x\ z + \(a\_\(1, 1, 0\)\) x\ y + a\_\(0, 0, 0\), {x, y, z}]\  // 
      MatrixForm\)], "Input",
  CellLabel->"In[6]:= ",
  CellTags->"CoefficientList"],

Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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                    {\(a\_\(0, 0, 1\)\)}
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              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {"0"},
                    {"0"}
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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                    {\(a\_\(1, 1, 1\)\)}
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              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {"0"},
                    {\(a\_\(1, 2, 1\)\)}
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {"0"},
                    {"0"}
                    }], "\[NoBreak]", ")"}], 
              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {\(a\_\(2, 2, 0\)\)},
                    {"0"}
                    }], "\[NoBreak]", ")"}]},
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {"0"},
                    {"0"}
                    }], "\[NoBreak]", ")"}]}
            }], "\[NoBreak]", ")"}],
      MatrixForm[ {{{
        Subscript[ a, 0, 0, 0], 
        Subscript[ a, 0, 0, 1]}, {0, 0}, {0, 0}}, {{0, 
        Subscript[ a, 1, 0, 1]}, {
        Subscript[ a, 1, 1, 0], 
        Subscript[ a, 1, 1, 1]}, {0, 
        Subscript[ a, 1, 2, 1]}}, {{0, 
        Subscript[ a, 2, 0, 1]}, {0, 0}, {
        Subscript[ a, 2, 2, 0], 0}}, {{0, 
        Subscript[ a, 3, 0, 1]}, {0, 0}, {0, 0}}}]]], "Output",
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  CellTags->"CoefficientList"]
}, Open  ]],

Cell["Collect", "ObjectName",
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Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Collect[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["x", "TI"],
  StyleBox["]", "MR"],
  " collects together terms involving the same powers of objects matching ",
  StyleBox["x", "TI"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Collect[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"x\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"x\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]", "MR"],
  " collects together terms that involve the same powers of objects matching \
",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"x\\\\\\\"\\\", \\\"TI\
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  ", ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"x\\\\\\\"\\\", \\\"TI\
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  ", \[Ellipsis] . \n\[FilledSmallSquare] ",
  StyleBox["Collect[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["var", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["h", "TI"],
  StyleBox["]", "MR"],
  " applies ",
  StyleBox["h", "TI"],
  " to the expression that forms the coefficient of each term obtained. "
}], "Usage",
  CellTags->{"S0.1075", "Collect"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Collect[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["x", "TI"],
  StyleBox["]", "MR"],
  " effectively writes ",
  StyleBox["expr", "TI"],
  " as a polynomial in ",
  StyleBox["x", "TI"],
  " or a fractional power of ",
  StyleBox["x", "TI"],
  ". \n\[FilledSmallSquare] Examples: ",
  StyleBox["Collect[x + n x + m, x]\[LongRightArrow]", "MR"],
  Cell[BoxData[
      \(m + \((1 + n)\)\ x\)], "InlineOutput"],
  "; ",
  StyleBox["Collect[(1+x+y)^3, x]\[LongRightArrow]", "MR"],
  Cell[BoxData[
      \(1 + x\^3 + 3\ y + 3\ y\^2 + y\^3 + x\^2\ \((3 + 3\ y)\) + 
        x\ \((3 + 6\ y + 3\ y\^2)\)\)], "InlineOutput"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Collect[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["x", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["Simplify]", "MR"],
  " can be used to simplify each coefficient separately. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.4.5",
    ButtonData:>{"1.4.5", "5.12"},
    ButtonStyle->"MainBookLink"],
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  ButtonBox["Section 3.3.1",
    ButtonData:>{"3.3.1", "1.2"},
    ButtonStyle->"MainBookLink"],
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    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["CoefficientList",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Together",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Cases",
    ButtonStyle->"RefGuideLink"],
  ". "
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  CellTags->{"S0.1075", "Collect"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"Collect"],

Cell[TextData[{
  "These two inputs collect terms first in the variable ",
  StyleBox["x", "MR"],
  " and then in the variable ",
  StyleBox["y", "MR"],
  ". "
}], "ExampleText",
  CellTags->"Collect"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Collect[x + 4\ y + 5\ x\ y, x]\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"Collect"],

Cell[BoxData[
    \(4\ y + x\ \((1 + 5\ y)\)\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"Collect"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Collect[x + 4\ y + 5\ x\ y, y]\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"Collect"],

Cell[BoxData[
    \(x + \((4 + 5\ x)\)\ y\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"Collect"]
}, Open  ]],

Cell[TextData[{
  "This first collects with respect to ",
  StyleBox["x", "MR"],
  ", then collects coefficients with respect to ",
  StyleBox["y", "MR"],
  " inside of each of the ",
  StyleBox["x", "MR"],
  " coefficients. "
}], "ExampleText",
  CellTags->"Collect"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Collect[
      x\^4 + \ \((a + y)\) \((x + y\^3)\) x\^2 + \((\ y\^3 + 3)\) x\^2, {x, 
        y}]\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"Collect"],

Cell[BoxData[
    \(x\^4 + x\^3\ \((a + y)\) + 
      x\^2\ \((3 + \((1 + a)\)\ y\^3 + y\^4)\)\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"Collect"]
}, Open  ]],

Cell[TextData[{
  "In this example we use ",
  StyleBox["Collect", "MR"],
  " to rewrite an expression by pulling out factors common only to certain \
parts. "
}], "ExampleText",
  CellTags->"Collect"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Collect[
      a\^2\ w\^2\  - \ a\^2\ x\  + \ \n\ \ 
        k1\ \((Exp[a\ c]\  + \ Exp[b\ c])\)\  + \ \n\ \ 
        k2\ \((Exp[a\ c]\  + \ Exp[b\ c])\), \n\ \ {a, \ \((Exp[a\ c]\  + \ 
            Exp[b\ c])\)}]\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"Collect"],

Cell[BoxData[
    \(\((\[ExponentialE]\^\(a\ c\) + \[ExponentialE]\^\(b\ c\))\)\ \((k1 + 
            k2)\) + a\^2\ \((w\^2 - x)\)\)], "Output",
  CellLabel->"Out[4]= ",
  CellTags->"Collect"]
}, Open  ]],

Cell["This example has the usual two arguments. ", "ExampleText",
  CellTags->"Collect"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Collect[
      Expand[\(\((\(-5\) + x)\)\^2\) \((2  x\^2 + 3  x - 2)\) 
            z + \(\((x\^2 - y\^2)\)\^2\) z\^3], z]\)], "Input",
  CellLabel->"In[5]:= ",
  CellTags->"Collect"],

Cell[BoxData[
    \(\((\(-50\) + 95\ x + 18\ x\^2 - 17\ x\^3 + 2\ x\^4)\)\ z + \((x\^4 - 
            2\ x\^2\ y\^2 + y\^4)\)\ z\^3\)], "Output",
  CellLabel->"Out[5]= ",
  CellTags->"Collect"]
}, Open  ]],

Cell[TextData[{
  StyleBox["Collect", "MR"],
  " applies an optional third argument to the innermost coefficients of the \
collected terms. "
}], "ExampleText",
  CellTags->"Collect"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell["\<\
This gives the continued fraction representation of a quadratic \
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
This gives the first few terms of the continued fraction \
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\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[TextData[{
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[9]= ",
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Cell[BoxData[
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Cell[TextData[{
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Sometimes the derivative is kept unevaluated until an argument is \
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[TextData[{
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Cell["Further Examples", "ExampleSection",
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Cell["\<\
This draws a circle with dashes three times as long as the spaces \
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Cell["Date", "ObjectName",
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Cell[TextData[{
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  " ",
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Cell[TextData[{
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\[FilledSmallSquare] ",
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mean time to obtain local time. \n\[FilledSmallSquare] All values returned by \
",
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  ", ",
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Cell["DeclarePackage", "ObjectName",
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Cell[TextData[{
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  Cell[BoxData[
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"name\\\\\\\"\\\", \
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    ButtonStyle->"MainBookLink"],
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Cell["Decompose", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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polynomials ",
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\[FilledSmallSquare] See ",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell["\<\
This polynomial can be rewritten in a variable that is a polynomial \
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\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
Here is a more complicated example. We first set up a triple \
composition. \
\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
The functions in the decomposition are different from the three we \
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\>", \
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
Recomposing the three new polynomials brings us back to the \
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\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["Decrement", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["DedekindEta", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  " is the ratio of Weierstrass half\[Hyphen]periods ",
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  StyleBox["FlattenAt", "MR"],
  " to the expression at that point. \n\[FilledSmallSquare] Example: ",
  StyleBox["Delete[", "MR"],
  Cell[BoxData[
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  StyleBox["a, ", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["b", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[", c", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[", ", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["2, 0", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]\[LongRightArrow]", "MR"],
  Cell[BoxData[
      \({a, b, c}\)], "InlineOutput"],
  ". \n\[FilledSmallSquare] Deleting the head of a whole expression makes the \
head be ",
  StyleBox["Sequence", "MR"],
  ". \n\[FilledSmallSquare] Example: ",
  StyleBox["Delete[", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["a, b", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[", 0]\[LongRightArrow]", "MR"],
  Cell[BoxData[
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  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.8.6",
    ButtonData:>{"1.8.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Insert",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["MapAt",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ReplacePart",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FlattenAt",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["DeleteCases",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Drop",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["StringDrop",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1114", "Delete"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"Delete"],

Cell["This deletes the third element from the list. ", "ExampleText",
  CellTags->"Delete"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"Delete"],

Cell[BoxData[
    \({a, b, d}\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"Delete"]
}, Open  ]],

Cell["DeleteCases", "ObjectName",
  CellTags->{"S0.1115", "DeleteCases"}],

Cell[TextData[{
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  StyleBox["DeleteCases[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["pattern", "TI"],
  StyleBox["]", "MR"],
  " removes all elements of ",
  StyleBox["expr", "TI"],
  " which match ",
  StyleBox["pattern", "TI"],
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  StyleBox["DeleteCases[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["pattern", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["levspec", "TI"],
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  StyleBox["expr", "TI"],
  " on levels specified by ",
  StyleBox["levspec", "TI"],
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  StyleBox["pattern", "TI"],
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}], "Usage",
  CellTags->{"S0.1115", "DeleteCases"}],

Cell[TextData[{
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  Cell[BoxData[
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  StyleBox[", _Integer]\[LongRightArrow]", "MR"],
  Cell[BoxData[
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  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  ", you can delete heads with ",
  StyleBox["DeleteCases", "MR"],
  ". Deleting the head of a particular element in an expression is equivalent \
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  StyleBox["FlattenAt", "MR"],
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  StyleBox["DeleteCases[", "MR"],
  Cell[BoxData[
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  StyleBox["1, f[2, 3], 4", "MR"],
  Cell[BoxData[
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  StyleBox[", f, ", "MR"],
  Cell[BoxData[
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  StyleBox["2", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[", Heads -> True]\[LongRightArrow]", "MR"],
  Cell[BoxData[
      \({1, 2, 3, 4}\)], "InlineOutput"],
  ". \n\[FilledSmallSquare] Level specifications are described in Section\
\[NonBreakingSpace]",
  ButtonBox["A.3.6",
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  ButtonBox["Section 2.3.2",
    ButtonData:>{"2.3.2", "2.12"},
    ButtonStyle->"MainBookLink"],
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  ButtonBox["Cases",
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  ", ",
  ButtonBox["ReplaceAll",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Delete",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1115", "DeleteCases"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"DeleteCases"],

Cell["\<\
This deletes all elements of the list that are real numbers. \
\>", \
"ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"DeleteCases"],

Cell[BoxData[
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  CellLabel->"Out[1]= ",
  CellTags->"DeleteCases"]
}, Open  ]],

Cell["This command deletes all integers from levels 1 to 2. ", "ExampleText",
  CellTags->"DeleteCases"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"DeleteCases"],

Cell[BoxData[
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  CellLabel->"Out[2]= ",
  CellTags->"DeleteCases"]
}, Open  ]],

Cell["\<\
In contrast this command deletes all integers only at level 2. \
\
\>", "ExampleText",
  CellTags->"DeleteCases"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[3]:= ",
  CellTags->"DeleteCases"],

Cell[BoxData[
    \({{2.`}, 3, 0.4`}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"DeleteCases"]
}, Open  ]],

Cell["DeleteDirectory", "ObjectName",
  CellTags->{"S0.1116", "DeleteDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["DeleteDirectory[\"", "MR"],
  StyleBox["dir", "TI"],
  StyleBox["\"]", "MR"],
  " deletes the specified directory. "
}], "Usage",
  CellTags->{"S0.1116", "DeleteDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["DeleteDirectory[\"", "MR"],
  StyleBox["dir", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["DeleteContents", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True]", "MR"],
  " deletes ",
  StyleBox["dir", "TI"],
  " and all files and directories that it contains. \n\[FilledSmallSquare] ",
  StyleBox["DeleteDirectory[\"", "MR"],
  StyleBox["dir", "TI"],
  StyleBox["\"]", "MR"],
  " deletes the directory ",
  StyleBox["dir", "TI"],
  " only if it contains no files. \n\[FilledSmallSquare] ",
  StyleBox["DeleteDirectory", "MR"],
  " returns ",
  StyleBox["Null", "MR"],
  " if it succeeds in deleting a directory, and ",
  StyleBox["$Failed", "MR"],
  " if it fails. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.6",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["CreateDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["DeleteFile",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1116", "DeleteDirectory"}],

Cell["DeleteFile", "ObjectName",
  CellTags->{"S0.1117", "DeleteFile"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["DeleteFile[\"", "MR"],
  StyleBox["file", "TI"],
  StyleBox["\"]", "MR"],
  " deletes a file. \n\[FilledSmallSquare] ",
  StyleBox["DeleteFile[", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"file\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"file\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox["\",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]", "MR"],
  " deletes a list of files. "
}], "Usage",
  CellTags->{"S0.1117", "DeleteFile"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["DeleteFile", "MR"],
  " returns ",
  StyleBox["Null", "MR"],
  " if it succeeds in deleting files, and ",
  StyleBox["$Failed", "MR"],
  " if it fails. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.11.2",
    ButtonData:>{"1.11.2", "2.3"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.11.6",
    ButtonData:>{"2.11.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["RenameFile",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["DeleteDirectory",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1117", "DeleteFile"}],

Cell["DelimiterFlashTime", "ObjectName",
  CellTags->{"S0.1117", "DelimiterFlashTime"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["DelimiterFlashTime", "MR"],
  " is an option for cells and notebooks which specifies how long in seconds \
a delimiter should flash for when its matching delimiter is entered. "
}], "Usage",
  CellTags->{"S0.1117", "DelimiterFlashTime"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["DelimiterFlashTime->0", "MR"],
  " makes delimiters not flash. \n\[FilledSmallSquare] A typical default \
setting is ",
  StyleBox["DelimiterFlashTime->0.3", "MR"],
  ", which makes matching delimiters flash for 0.3 seconds. \n\
\[FilledSmallSquare] Delimiters include parentheses, brackets and braces, as \
well as ",
  StyleBox["[[", "MR"],
  ", ",
  StyleBox["]]", "MR"],
  " and ",
  StyleBox["(*", "MR"],
  " and ",
  StyleBox["*)", "MR"],
  ", and paired special characters such as ",
  Cell[BoxData[
      \(TraditionalForm\`\[LeftAngleBracket]\)], "InlineFormula"],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\[RightAngleBracket]\)], "InlineFormula"],
  ". \n\[FilledSmallSquare] If you enter an unpaired closing delimiter the \
standard ",
  StyleBox["Mathematica", "TI"],
  " front end will beep. \n\[FilledSmallSquare] You can use the front end \
menu item ",
  StyleBox["Check Balance", "SR"],
  " to select ranges with balanced delimiters in an expressions. \n\
\[FilledSmallSquare] You can set ",
  StyleBox["DelimiterFlashTime", "MR"],
  " at the level of a single cell, a notebook, or the whole front end. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.11",
    ButtonData:>{"2.10.11", "11.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["ShowAutoStyles",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["SyntaxQ",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ShowCursorTracker",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1117", "DelimiterFlashTime"}],

Cell["Denominator", "ObjectName",
  CellTags->{"S0.1118", "Denominator"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Denominator[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " gives the denominator of ",
  StyleBox["expr", "TI"],
  ". "
}], "Usage",
  CellTags->{"S0.1118", "Denominator"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Denominator", "MR"],
  " picks out terms which have superficially negative exponents. ",
  StyleBox["Numerator", "MR"],
  " picks out all remaining terms. \n\[FilledSmallSquare] An exponent is \
\[OpenCurlyDoubleQuote]superficially negative\[CloseCurlyDoubleQuote] if it \
has a negative number as a factor. \n\[FilledSmallSquare] The standard \
representation of rational expressions as products of powers means that you \
cannot simply use ",
  StyleBox["Part", "MR"],
  " to extract denominators. \n\[FilledSmallSquare] ",
  StyleBox["Denominator", "MR"],
  " can be used on rational numbers. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.4.7",
    ButtonData:>{"1.4.7", "7.9"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["ExpandDenominator",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Rationals",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1118", "Denominator"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"Denominator"],

Cell["Here is a numerator and denominator. ", "ExampleText",
  CellTags->"Denominator"],

Cell[CellGroupData[{

Cell[BoxData[
    \({Numerator[\(\(\[ExponentialE]\^\(-a\)\) \[ExponentialE]\^b\)\/\(\(\
\[ExponentialE]\^\(-c\)\) \[ExponentialE]\^d\)], 
      Denominator[\(\(\[ExponentialE]\^\(-a\)\) \[ExponentialE]\^b\)\/\(\(\
\[ExponentialE]\^\(-c\)\) \[ExponentialE]\^d\)]}\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"Denominator"],

Cell[BoxData[
    \({\[ExponentialE]\^\(b + c\), \[ExponentialE]\^\(a + d\)}\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"Denominator"]
}, Open  ]],

Cell["DensityGraphics", "ObjectName",
  CellTags->{"S0.1119", "DensityGraphics"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["DensityGraphics[", "MR"],
  StyleBox["array", "TI"],
  StyleBox["]", "MR"],
  " is a representation of a density plot. "
}], "Usage",
  CellTags->{"S0.1119", "DensityGraphics"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["array", "TI"],
  " must be a rectangular array of real numbers, representing ",
  Cell[BoxData[
      \(TraditionalForm\`z\)], "InlineFormula"],
  " values. \n\[FilledSmallSquare] The following options can be given: "
}], "Notes",
  CellTags->{"S0.1119", "DensityGraphics"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"AspectRatio\\\\\\\"\\\", \\\"MR\
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\\) \"\>", "\<\"ratio of height to width \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Axes\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"False\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"whether to draw axes \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"AxesLabel\\\\\\\"\\\", \\\"MR\\\
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\\\"MR\\\"]\\) \"\>", "\<\"axes labels \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"AxesOrigin\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", "\<\"where axes should cross \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"AxesStyle\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \("graphics directives to specify the style for axes \
"\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Background\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", "\<\"background color for the plot \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"ColorFunction\\\\\\\"\\\", \
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"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"ColorFunctionScaling\\\\\\\"\\\"\
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\\\"MR\\\"]\\) \"\>", 
            RowBox[{"\<\"whether to scale \"\>", 
              StyleBox["z",
                
                "InlineFormula"], "\<\" values before applying a color \
function \"\>"}]},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"ColorOutput\\\\\\\"\\\", \\\"MR\
\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"type of color output to produce \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"DefaultColor\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"DisplayFunction\\\\\\\"\\\", \\\
\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"$DisplayFunction\\\\\\\
\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"function for generating output \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Epilog\\\\\\\"\\\", \\\"MR\\\"]\
\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"{}\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", \("graphics primitives to be rendered after the main plot "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"FormatType\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"$FormatType\\\\\\\"\\\", \\\"MR\\\"]\\) \"\
\>", "\<\"the default format type for text \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Frame\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\
\"MR\\\"]\\) \"\>", "\<\"whether to put a frame around the plot \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"FrameLabel\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"None\\\\\\\"\\\", \\\
\"MR\\\"]\\) \"\>", "\<\"frame labels \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"FrameStyle\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", \("graphics directives giving the style for the frame "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"FrameTicks\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", "\<\"frame tick marks \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"ImageSize\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \("the absolute size at which to render the graphic in \
a notebook "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Mesh\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\
\"MR\\\"]\\) \"\>", "\<\"whether to draw a mesh \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"MeshRange\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", 
            RowBox[{"\<\"ranges of \"\>", 
              StyleBox["x",
                "InlineFormula"], "\<\" and \"\>", 
              StyleBox["y",
                "InlineFormula"], "\<\" coordinates \"\>"}]},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"MeshStyle\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \("graphics directives to specify the style for mesh \
lines "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"PlotLabel\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"None\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"a label for the plot \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"PlotRange\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", 
            RowBox[{"\<\"range of \"\>", 
              StyleBox["z",
                "InlineFormula"], "\<\" values to include \"\>"}]},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"PlotRegion\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", "\<\"the final display region to be filled \"\>"}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"DensityGraphics"],

Cell[BoxData[
    FormBox[GridBox[{
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Prolog\\\\\\\"\\\", \\\"MR\\\"]\
\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"{}\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", \("graphics primitives to be rendered before the main plot "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"RotateLabel\\\\\\\"\\\", \\\"MR\
\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", 
            RowBox[{"\<\"whether to rotate \"\>", 
              StyleBox["y",
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The singularities of this function show up as sharp changes in \
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This is the determinant of the same matrix over the integers modulo \
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This computes the determinant of the matrix and again times the \
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Here is its determinant with how long the calculation takes. \
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Cell["\<\
This displays the result in traditional matrix form, instead of as \
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Cell["DigitBlock", "ObjectName",
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This expresses the classic distributive law of multiplication over \
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Here we distribute over a product instead of a sum (two \
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"MGF\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PBM\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"portable bitmap format \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PGM\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"portable graymap format \
(\\!\\(\\*StyleBox[\\\"\\\\\\\".pgm\\\\\\\"\\\", \\\"MR\\\"]\\)) \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PNG\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"PNG format (\\!\\(\\*StyleBox[\\\
\"\\\\\\\".png\\\\\\\"\\\", \\\"MR\\\"]\\)) \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PNM\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"portable anymap format \
(\\!\\(\\*StyleBox[\\\"\\\\\\\".pnm\\\\\\\"\\\", \\\"MR\\\"]\\)) \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PPM\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"portable pixmap format \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PSImage\\\\\\\\\
\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"PostScript \
\\!\\(\\*StyleBox[\\\"\\\\\\\"image\\\\\\\"\\\", \\\"MR\\\"]\\) format \
(\\!\\(\\*StyleBox[\\\"\\\\\\\".psi\\\\\\\"\\\", \\\"MR\\\"]\\)) \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"TIFF\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"TIFF (\\!\\(\\*StyleBox[\\\"\\\\\
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\\\"\\\", \\\"MR\\\"]\\)) \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"XBitmap\\\\\\\\\
\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"X window system bitmap \
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "2ColumnTable",
  CellTags->"Export"],

Cell["\<\
\[FilledSmallSquare] The following sound formats are supported: \
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\>", "Notes",
  CellTags->{"S0.1174", "Export"}],

Cell[BoxData[
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                "InlineFormula"], "\<\" law encoding \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"WAV\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"Microsoft wave format \
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "2ColumnTable",
  CellTags->"Export"],

Cell["\<\
\[FilledSmallSquare] The following general options can be given: \
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\>", "Notes",
  CellTags->{"S0.1174", "Export"}],

Cell[BoxData[
    FormBox[GridBox[{
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"ByteOrdering\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"$ByteOrdering\\\\\\\"\\\", \\\"MR\\\"]\\) \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"CharacterEncoding\\\\\\\"\\\", \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"ConversionOptions\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"{}\\\\\\\"\\\", \
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Export"],

Cell[TextData[{
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  StyleBox["Cell", "MR"],
  " objects, as well as any box expression obtained from ",
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  ", can be exported in the following formats: "
}], "Notes",
  CellTags->{"S0.1174", "Export"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"TeX\\\\\\\\\\\\\
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "2ColumnTable",
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Cell[TextData[{
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  StyleBox["Export", "MR"],
  " operates like ",
  StyleBox["HTMLSave", "MR"],
  " and ",
  StyleBox["TeXSave", "MR"],
  ". \n\[FilledSmallSquare] Many details can be specified in the setting for \
",
  StyleBox["ConversionOptions", "MR"],
  ". \n\[FilledSmallSquare] Possible formats accepted by ",
  StyleBox["Export", "MR"],
  " are given in the list ",
  StyleBox["$ExportFormats", "MR"],
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  StyleBox["Export[\"!", "MR"],
  StyleBox["prog", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["\"", "MR"],
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  ButtonBox["ExportString",
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  ButtonBox["Display",
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  ", ",
  ButtonBox["Write",
    ButtonStyle->"RefGuideLink"],
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    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1174", "Export"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Additional Information", "ExampleSection",
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Cell[TextData[{
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  StyleBox["Export", "MR"],
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  StyleBox["ConversionOptions", "MR"],
  ": ",
  StyleBox["Export[\"", "MR"],
  StyleBox["file", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["format", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["ConversionOptions", "MR"],
  " ",
  StyleBox["-> ", "MR"],
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  Cell[BoxData[
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  StyleBox["]", "MR"],
  ". "
}], "Text",
  CellTags->"Export"],

Cell[TextData[StyleBox["AIFF", "MB"]], "Subsubsection",
  CellTags->"Export"],

Cell["\<\
\[FilledSmallSquare] The following option can be given: \
\>", \
"Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[BoxData[
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Export"],

Cell[TextData[{
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  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["8", "MR"],
  " unless a different depth is explicitly specified in the ",
  StyleBox["Sound", "MR"],
  " object. ",
  StyleBox["Sound", "MR"],
  " can be exported to \"AIFF\" with a ",
  StyleBox["\"SampleDepth\"", "MR"],
  " of ",
  StyleBox["8", "MR"],
  " or ",
  StyleBox["16", "MR"],
  ". "
}], "Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[TextData[StyleBox["AU", "MB"]], "Subsubsection",
  CellTags->"Export"],

Cell["\<\
\[FilledSmallSquare] The following option can be given: \
\>", \
"Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"SampleDepth\\\\\
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\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"sample depth of sound \
in bits \"\>"}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Export"],

Cell[TextData[{
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  StyleBox["->", "MR"],
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  StyleBox["8", "MR"],
  " which creates a mu\[Hyphen]law 8\[Hyphen]bit encoded sound file, unless a \
different depth is explicitly specified in the ",
  StyleBox["Sound", "MR"],
  " object. ",
  StyleBox["Sound", "MR"],
  " can be exported to \"AU\" with a ",
  StyleBox["\"SampleDepth\"", "MR"],
  " of ",
  StyleBox["8", "MR"],
  ", ",
  StyleBox["16", "MR"],
  " or ",
  StyleBox["32", "MR"],
  ". "
}], "Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[TextData[StyleBox["GIF", "MB"]], "Subsubsection",
  CellTags->"Export"],

Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
"Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[BoxData[
    FormBox[GridBox[{
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Background\\\\\\\
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\\\"False\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \("sets the background color of \
the image "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
ColorReductionDither\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\
\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Interlace\\\\\\\
\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Transparency\\\\\
\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\
\\\\\"False\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \("sets the transparent color \
of the image "\)}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Export"],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["\"Background\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["RGBColor[", "MR"],
  StyleBox["r", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["g", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["b", "TI"],
  StyleBox["]", "MR"],
  " sets the background color of the exported GIF to the specified graphics \
directive. Many applications will ignore this setting and will also not \
display this color if there is no transparency set. \n\[FilledSmallSquare] ",
  StyleBox["\"ColorReductionDither\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  " will provide better resulting images once a color map is chosen by \
dithering the image. \n\[FilledSmallSquare] ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["Automatic", "MR"],
  " will choose an adaptive set of at most ",
  StyleBox["256", "MR"],
  " colors. ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["n", "TI"],
  " will choose at most ",
  StyleBox["n", "TI"],
  " colors for the color palette. ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"color\\\\\\\"\\\", \\\
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  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"color\\\\\\\"\\\", \\\
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  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  " specifies a list of color directives to use to color reduce the image. \
The values for ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"color\\\\\\\"\\\", \\\
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  ", ",
  StyleBox["GrayLevel", "MR"],
  ", ",
  StyleBox["Hue", "MR"],
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  StyleBox["RGBColor", "MR"],
  " directives. The number of colors chosen for ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " must be between ",
  StyleBox["2", "MR"],
  " and ",
  StyleBox["256", "MR"],
  ". \n\[FilledSmallSquare] The following example produces a color map of \
\"browser safe\" ",
  StyleBox["216", "MR"],
  " colors: ",
  StyleBox["Export[\"file.gif\",", "MR"],
  " ",
  StyleBox["gr,", "MR"],
  " ",
  StyleBox["\"GIF\",", "MR"],
  " ",
  StyleBox["ConversionOptions", "MR"],
  " ",
  StyleBox["-> {\"ColorReductionPalette\"", "MR"],
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  StyleBox["->", "MR"],
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  StyleBox["Table[RGBColor[i,j,k],", "MR"],
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  StyleBox["{i,0,1,0.2},{j,0,1,0.2},{k,0,1,0.2}]", "MR"],
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  StyleBox["}]", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["\"Transparency\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["RGBColor[", "MR"],
  StyleBox["r", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["g", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["b", "TI"],
  StyleBox["]", "MR"],
  " requests that the specified color directive defines the transparent color \
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  StyleBox["\"Transparency\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["GrayLevel[1]", "MR"],
  " defines white as the transparent color. By default an adaptive ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " is used to produce the GIF color palette, and this could in some cases \
produce a set of colors which does not contain the transparent color \
requested. \n\[FilledSmallSquare] A list of graphics objects exported to the \
GIF format will produce an animated GIF. \n\[FilledSmallSquare] The following \
additional options can be given when exporting as an animated GIF: "
}], "Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[BoxData[
    FormBox[GridBox[{
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
AnimationDisplayTime\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Disposal\\\\\\\\\
\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\
\"None\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \("determines how each frame is \
replaced when displayed "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
GlobalColorReduction\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\
\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
\("specifies whether to create one color palette shared by all frames "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Loop\\\\\\\\\\\\\
\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"False\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
\("determines whether an animation loops when finished "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Pause\\\\\\\\\\\
\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"False\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
\("determines whether there is a pause after each frame "\)}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Export"],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["\"AnimationDisplayTime\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["n", "TI"],
  " sets the delay time in seconds for each frame. By default no delay time \
is used. If a list of delay times is used, the sequence of delay times is \
applied to the entire list of frames, repeating the delay time sequence if \
necessary. \n\[FilledSmallSquare] The option ",
  StyleBox["\"Disposal\"", "MR"],
  " tells an application what should replace each frame after the frame is \
done displaying. The option can have the values of ",
  StyleBox["None", "MR"],
  ", ",
  StyleBox["False", "MR"],
  ", ",
  StyleBox["Background", "MR"],
  ", or ",
  StyleBox["Previous", "MR"],
  ". The default setting of ",
  StyleBox["\"Disposal\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["None", "MR"],
  " will not do anything, and with no transparency in the GIF, this setting \
will have no noticeable effect. However, if you are creating a transparent \
GIF animation, you will usually want ",
  StyleBox["\"Disposal\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["Background", "MR"],
  " as this will clear each frame in the animation before the next frame is \
displayed, otherwise you will see each frame image drawn on top of the other. \
\n\[FilledSmallSquare] When exporting a list of images for an animated GIF, \
",
  StyleBox["\"GlobalColorReduction\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  " will build one color reduction palette from all images to get one global \
color map used by the entire GIF, as opposed to building a unique color map \
for each frame which could increase the size of the resulting animated GIF. \n\
\[FilledSmallSquare] ",
  StyleBox["\"Loop\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  " will create an animated GIF that will loop forever, while ",
  StyleBox["\"Loop\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["n", "TI"],
  " will make the animation loop ",
  StyleBox["n", "TI"],
  " times. \n\[FilledSmallSquare] With the setting ",
  StyleBox["\"Pause\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  " user intervention is required for the animated GIF to continue to the \
next frame. Currently, there are very few applications which support this \
feature of the GIF format. "
}], "Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[TextData[StyleBox["HDF", "MB"]], "Subsubsection",
  CellTags->"Export"],

Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
"Notes",
  CellTags->{"Export", "S0.1174"}],

Cell[BoxData[
    FormBox[GridBox[{
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Append\\\\\\\\\\\
\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"False\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
\("specifies whether to add the data to an existing file "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"FormatType\\\\\\\
\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Interface\\\\\\\
\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\
\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \("determines what HDF \
dataset interface is created "\)}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Export"],

Cell[TextData[{
  "\[FilledSmallSquare] With the setting ",
  StyleBox["\"Append\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  ", the HDF dataset is appended to the existing file instead of completely \
replacing any existing file. The file is created if it does not already \
exist. \n\[FilledSmallSquare] With the default setting of ",
  StyleBox["\"FormatType\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["Automatic", "MR"],
  " a multidimensional array of integers is exported as an array of 32\
\[Hyphen]bit integers, and real arrays are exported as an array of \
32\[Hyphen]bit reals. If an array contains a mixture of integers and reals, \
all data in the array will be exported as reals. All values are exported \
modulo the specified data format size limits. \n\[FilledSmallSquare] ",
  StyleBox["\"FormatType\"", "MR"],
  " can be set to the following possible binary formats: "
}], "Notes",
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\[FilledSmallSquare] The following options can be given: \
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\[FilledSmallSquare] The following options can be given: \
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whether the ",
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\[FilledSmallSquare] A list of graphics objects can be exported to the TIFF \
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applications will only read the first image of a TIFF file containing \
multiple images. "
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\[FilledSmallSquare] The following option can be given: \
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",
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  StyleBox["n", "TI"],
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
This picks out the element from the upper left corner of the matrix \
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Cell["It is not necessary to name the blanks. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
  "We repeat the last example over the integers modulo ",
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Cell[BoxData[
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Cell[BoxData[
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Cell["False", "ObjectName",
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Cell[TextData[{
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell["Evaluate the cells to compare the graphics. ", "Text",
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Cell[BoxData[
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Cell[BoxData[
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  CellTags->"False"],

Cell["Fibonacci", "ObjectName",
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Cell[TextData[{
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  StyleBox["]", "MR"],
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  StyleBox["Fibonacci[", "MR"],
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  StyleBox[",", "MR"],
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  StyleBox["x", "TI"],
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Cell[TextData[{
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  " with ",
  Cell[BoxData[
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  ". \n\[FilledSmallSquare] For any complex value of ",
  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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    ButtonStyle->"MainBookLink"],
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Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Fibonacci"],

Cell[BoxData[
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Cell[TextData[{
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Cell[BoxData[
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Cell[BoxData[
    \(434665576869374564356885276750406258025646605173717804024817290895365554\
179490518904038798400792551692959225930803226347752096896232398733224711616429\
96440906533187938298969649928516003704476137795166849228875\)], "Output",
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  "Here is an identity involving ",
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Cell[CellGroupData[{

Cell[BoxData[
    \(FullSimplify[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[4]= ",
  CellTags->"Fibonacci"]
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Cell[TextData[{
  "You can get rid of the ",
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  ". "
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[5]:= ",
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Cell[BoxData[
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  CellLabel->"Out[5]= ",
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Cell[TextData[{
  "Here is a similar example for ",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["FileByteCount", "ObjectName",
  CellTags->{"S0.1185", "FileByteCount"}],

Cell[TextData[{
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}], "Usage",
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Cell[TextData[{
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system to another, the number of bytes in the file as reported by ",
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  " may change. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.6",
    ButtonData:>{"2.11.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["StringLength",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FileType",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
  ButtonBox["Utilities`BinaryFiles`",
    ButtonData:>{"Utilities`BinaryFiles`"},
    ButtonStyle->"AddOnsLink"],
  ". "
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  CellTags->{"S0.1185", "FileByteCount"}],

Cell["FileDate", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["Date", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.6",
    ButtonData:>{"2.11.6", "6.2"},
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Cell["FileNames", "ObjectName",
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Cell[TextData[{
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",
  StyleBox["FileNames[\"", "MR"],
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  Cell[BoxData[
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  StyleBox["\"", "MR"],
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  StyleBox["\"", "MR"],
  Cell[BoxData[
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"form\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"form\\\\\\\"\\\", \
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  StyleBox["FileNames[", "MR"],
  StyleBox["forms", "TI"],
  StyleBox[",", "MR"],
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  Cell[BoxData[
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"dir\\\\\\\"\\\", \
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"dir\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
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  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"dir\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "i"], TraditionalForm]], "InlineFormula"],
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  StyleBox["forms", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["dirs", "TI"],
  StyleBox[",", "MR"],
  " ",
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Cell[TextData[{
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  StyleBox["\"", "MR"],
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  StyleBox["FileNames[", "MR"],
  StyleBox["forms", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["dirs", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["FileNames[", "MR"],
  StyleBox["forms", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["dirs", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["n", "TI"],
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names of files. \n\[FilledSmallSquare] Setting the option ",
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  " ",
  StyleBox["True", "MR"],
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MS\[Hyphen]DOS, ",
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as equivalent. \n\[FilledSmallSquare] See ",
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  ButtonBox["Section 1.11.2",
    ButtonData:>{"1.11.2", "2.3"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.11.4",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FileType",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Get",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
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    ButtonStyle->"AddOnsLink"],
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  CellTags->{"S0.1186", "FileNames"}],

Cell["FileType", "ObjectName",
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Cell[TextData[{
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  StyleBox["\"]", "MR"],
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  StyleBox["File", "MR"],
  ", ",
  StyleBox["Directory", "MR"],
  " or ",
  StyleBox["None", "MR"],
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}], "Usage",
  CellTags->{"S0.1187", "FileType"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["FileType", "MR"],
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  StyleBox["None", "MR"],
  " if the file specified does not exist. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.6",
    ButtonData:>{"2.11.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["FileNames",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FileByteCount",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1187", "FileType"}],

Cell["Find", "ObjectName",
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Cell[TextData[{
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  StyleBox["Find[", "MR"],
  StyleBox["stream", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["text", "TI"],
  StyleBox["\"]", "MR"],
  " finds the first line in an input stream that contains the specified \
string. \n\[FilledSmallSquare] ",
  StyleBox["Find[", "MR"],
  StyleBox["stream", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"text\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
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This finds a minimum, explicitly specifying a gradient to use. \
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\>", "ExampleText",
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Using the default setting, a different local minimum is found. \
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  StyleBox["lhs", "TI"],
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  StyleBox[",", "MR"],
  " ",
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  Cell[BoxData[
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  Cell[BoxData[
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  StyleBox["x", "TI"],
  ". This form must be used if symbolic derivatives of the equation cannot be \
found. \n\[FilledSmallSquare] ",
  StyleBox["FindRoot[", "MR"],
  StyleBox["lhs", "TI"],
  StyleBox["==", "MR"],
  StyleBox["rhs", "TI"],
  StyleBox[",", "MR"],
  " ",
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  StyleBox["x", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["xstart", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["xmin", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["xmax", "TI"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]", "MR"],
  " searches for a solution, stopping the search if ",
  StyleBox["x", "TI"],
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  StyleBox["xmin", "TI"],
  " to ",
  StyleBox["xmax", "TI"],
  ". \n\[FilledSmallSquare] ",
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  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"eqn\\\\\\\"\\\", \
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      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"eqn\\\\\\\"\\\", \
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  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
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  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["x", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
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  Cell[BoxData[
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  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["y", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"y\\\\\\\"\\\", \\\"TI\
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  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  StyleBox["]", "MR"],
  " searches for a numerical solution to the simultaneous equations ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"eqn\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "i"], TraditionalForm]], "InlineFormula"],
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  StyleBox["FindRoot", "MR"],
  " returns a list of replacements for ",
  StyleBox["x", "TI"],
  ", ",
  StyleBox["y", "TI"],
  ", \[Ellipsis] , in the same form as obtained from ",
  StyleBox["Solve", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["FindRoot", "MR"],
  " has attribute ",
  StyleBox["HoldAll", "MR"],
  ". \n\[FilledSmallSquare] If you specify only one starting value of ",
  StyleBox["x", "TI"],
  ", ",
  StyleBox["FindRoot", "MR"],
  " searches for a solution using Newton's method. If you specify two \
starting values, ",
  StyleBox["FindRoot", "MR"],
  " uses a variant of the secant method. \n\[FilledSmallSquare] If all \
equations and starting values are real, then ",
  StyleBox["FindRoot", "MR"],
  " will search only for real roots. If any are complex, it will also search \
for complex roots. \n\[FilledSmallSquare] You can always tell ",
  StyleBox["FindRoot", "MR"],
  " to search for complex roots by adding ",
  StyleBox["0.", "MR"],
  " ",
  StyleBox["I", "MR"],
  " to the starting value. \n\[FilledSmallSquare] The following options can \
be given: "
}], "Notes",
  CellTags->{"S0.1191", "FindRoot"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"AccuracyGoal\\\\\\\"\\\", \
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\"\>", \("the accuracy sought in the value of the function at the root "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Compiled\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"DampingFactor\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"1\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"damping factor in Newton's method \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Jacobian\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", "\<\"the Jacobian of the system \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"MaxIterations\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"15\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"maximum number of iterations used \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"WorkingPrecision\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"$MachinePrecision\\\\\\\"\\\", \\\"MR\\\"]\
\\) \"\>", \("the number of digits used in internal computations "\)}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"FindRoot"],

Cell[TextData[{
  "\[FilledSmallSquare] The default setting for ",
  StyleBox["AccuracyGoal", "MR"],
  " is 10 digits less than ",
  StyleBox["WorkingPrecision", "MR"],
  ". \n\[FilledSmallSquare] If ",
  StyleBox["FindRoot", "MR"],
  " does not succeed in finding a solution to the accuracy you specify within \
",
  StyleBox["MaxIterations", "MR"],
  " steps, it returns the most recent approximation to a solution that it \
found. You can then apply ",
  StyleBox["FindRoot", "MR"],
  " again, with this approximation as a starting point. \n\
\[FilledSmallSquare] The size of each step taken in Newton's method is \
multiplied by the setting given for ",
  StyleBox["DampingFactor", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.5.7",
    ButtonData:>{"1.5.7", "7.17"},
    ButtonStyle->"MainBookLink"],
  ", ",
  ButtonBox["Section 1.6.3",
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    ButtonStyle->"MainBookLink"],
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  ButtonBox["Section 3.9.6",
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    ButtonStyle->"MainBookLink"],
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    ButtonStyle->"MainBookLink"],
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  ", ",
  ButtonBox["Solve",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FindMinimum",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
  ButtonBox["NumericalMath`InterpolateRoot`",
    ButtonData:>{"NumericalMath`InterpolateRoot`"},
    ButtonStyle->"AddOnsLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1191", "FindRoot"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"FindRoot"],

Cell["These two curves intersect at one point. ", "ExampleText",
  CellTags->"FindRoot"],

Cell["Evaluate the cell to see the graphic. ", "Text",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(Plot[{Exp[x], x\^2}, {x, \(-1\), 1}]; \)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"FindRoot"],

Cell[TextData[{
  "This finds a numerical approximation to the ",
  StyleBox["x", "MR"],
  " coordinate of the intersection point. The ",
  Cell[BoxData[
      \(TraditionalForm\`\(-0.5\)\)], "InlineFormula"],
  " is the initial guess. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[Exp[x] == x\^2, {x, \(-0.5\)}]\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] \(-0.7034674295409824`\)}\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "Trigonometric equations typically have an infinite number of roots. If you \
start sufficiently close to a particular root of an equation, ",
  StyleBox["FindRoot", "MR"],
  " will find that root. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[Sin[x\^2] == 0, {x, 2}]\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 1.772453850905937`}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell["\<\
Starting closer to another root will give different solution. \
\
\>", "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[Sin[x\^2] == 0, {x, 3}]\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 3.069980123840709`}\)], "Output",
  CellLabel->"Out[4]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "You can restrict ",
  StyleBox["FindRoot", "MR"],
  " to a region in which to look for solutions. Here the initial guess is ",
  Cell[BoxData[
      \(TraditionalForm\`1\)], "InlineFormula"],
  " and the solution is supposed to be between ",
  Cell[BoxData[
      \(TraditionalForm\`0.5\)], "InlineFormula"],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`1.5\)], "InlineFormula"],
  ". There is no such solution. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[Sin[x\^2] == 0, {x, 1. ,  .5, 1.5}]\)], "Input",
  CellLabel->"In[5]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(FindRoot::"regex" \(\(:\)\(\ \)\) 
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  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 0.22129613767254896`}\)], "Output",
  CellLabel->"Out[5]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "This is what happens when ",
  StyleBox["FindRoot", "MR"],
  " can find no solutions at all. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[Sin[x\^2] == \[Pi], {x, 1}]\)], "Input",
  CellLabel->"In[6]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(FindRoot::"cvnwt" \(\(:\)\(\ \)\) 
      "Newton's method failed to converge to the prescribed accuracy after \!\
\(15\) iterations."\)], "Message",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 22.06298018275477`}\)], "Output",
  CellLabel->"Out[6]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "If you want ",
  StyleBox["FindRoot", "MR"],
  " to use complex values in its search, then you need to give a complex \
starting value. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[Sin[x\^2] == \[Pi], {x, 1 + \[ImaginaryI]}]\)], "Input",
  CellLabel->"In[7]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] \(\(1.4086357790876853`\)\(\[InvisibleSpace]\)\) + 
          0.6430073336777856`\ \[ImaginaryI]}\)], "Output",
  CellLabel->"Out[7]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "For this equation, ",
  StyleBox["FindRoot", "MR"],
  " has difficulty using Newton's method with only the default setting. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[\[ExponentialE]\^Abs[x] == 2, {x, 1}]\)], "Input",
  CellLabel->"In[8]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(FindRoot::"frjc" \(\(:\)\(\ \)\) 
      "Could not symbolically find the Jacobian of \
\!\({\[ExponentialE]\^\(Abs[x]\) - 2}\). Try giving two starting values for \
each variable."\)], "Message",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(FindRoot[\[ExponentialE]\^Abs[x] == 2, {x, 1}]\)], "Output",
  CellLabel->"Out[8]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell["\<\
When you specify the Jacobian (derivative), Newton's method can be \
used. \
\>", "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[\[ExponentialE]\^Abs[x] == 2, {x, 1}, 
      Jacobian \[Rule] {{Sign[x]\ \[ExponentialE]\^Abs[x]}}]\)], "Input",
  CellLabel->"In[9]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 0.6931475810597714`}\)], "Output",
  CellLabel->"Out[9]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell["\<\
When you don't know the Jacobian or it cannot be expressed by a \
formula, you can use the secant method by giving two starting values. \
\>", \
"ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[\[ExponentialE]\^Abs[x] == 2, {x, {0, 1}}]\)], "Input",
  CellLabel->"In[10]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 0.6931471804286731`}\)], "Output",
  CellLabel->"Out[10]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "Here is a way to count the number of times ",
  StyleBox["FindRoot", "MR"],
  " evaluates the function. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(n = 0; \)], "Input",
  CellLabel->"In[11]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
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        2, {x, {0, 1}}]; \)], "Input",
  CellLabel->"In[12]:= ",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(n\)], "Input",
  CellLabel->"In[13]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(7\)], "Output",
  CellLabel->"Out[13]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[BoxData[
    \(n =  . \)], "Input",
  CellLabel->"In[14]:= ",
  CellTags->"FindRoot"],

Cell["\<\
The advantage of Newton's method is that it usually converges \
faster and takes fewer steps than the secant method. \
\>", "ExampleText",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(n = 0; \)], "Input",
  CellLabel->"In[15]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(FindRoot[\(n++\); \[ExponentialE]\^Abs[x] == 2, {x, 1}, 
      Jacobian \[Rule] {{Sign[x]\ \[ExponentialE]\^Abs[x]}}]; \)], "Input",
  CellLabel->"In[16]:= ",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(n\)], "Input",
  CellLabel->"In[17]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(5\)], "Output",
  CellLabel->"Out[17]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[BoxData[
    \(n =  . \)], "Input",
  CellLabel->"In[18]:= ",
  CellTags->"FindRoot"],

Cell[TextData[{
  "This finds a solution to a set of simultaneous equations. It is a good \
idea to avoid taking the starting values for ",
  StyleBox["x", "MR"],
  " and ",
  StyleBox["y", "MR"],
  " to be equal or to take any other \"special\" combinations of values. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[{x\^2 + y\^2 == 1, 
        y == x\ Exp[x]}, {x,  .1}, {y,  .5}]\)], "Input",
  CellLabel->"In[19]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 0.5134886100441767`, 
      y \[Rule] 0.858096409132671`}\)], "Output",
  CellLabel->"Out[19]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell["\<\
If you specify the Jacobian in a problem with several variables, \
you need to make sure the rows and columns are ordered properly. \
\>", \
"ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[{x\^2 + Abs[y] - 3, x\ y + 1}, {x, 1}, {y, \(-1\)}, 
      Jacobian \[Rule] {{2\ x, Sign[y]}, {y, x}}]\)], "Input",
  CellLabel->"In[20]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 1.5320888862389288`, 
      y \[Rule] \(-0.6527036446641683`\)}\)], "Output",
  CellLabel->"Out[20]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "The criterion ",
  StyleBox["FindRoot", "MR"],
  " uses for convergence to a root is how close the function is to zero. If \
the function is very flat it may be close to zero before the variable is \
close enough to the root. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[x\^4, \ {x, \ 1}]\)], "Input",
  CellLabel->"In[21]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 0.023757264018058777`}\)], "Output",
  CellLabel->"Out[21]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "Here ",
  StyleBox["FindRoot", "MR"],
  " stopped when the value of the function was within the default ",
  StyleBox["AccuracyGoal", "MR"],
  " (6 digits) of zero, but the root is only good to one decimal place. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(x\^4 /. %\)], "Input",
  CellLabel->"In[22]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(3.185559317401524`*^-7\)], "Output",
  CellLabel->"Out[22]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "To get closer to the actual value of the root, increase the setting of the \
",
  StyleBox["AccuracyGoal", "MR"],
  " option. It is often also necessary to increase the setting of the ",
  StyleBox["WorkingPrecision", "MR"],
  " option. \nIn this case, ",
  StyleBox["FindRoot", "MR"],
  " does not make it to the specified ",
  StyleBox["AccuracyGoal", "MR"],
  ". The convergence to this root is very slow; we get ",
  Cell[BoxData[
      \(TraditionalForm\`3\)], "InlineFormula"],
  " decimal places. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[x\^4, {x, 0.0237573}, \n\tAccuracyGoal \[Rule] 24, 
      WorkingPrecision \[Rule] 34]\)], "Input",
  CellLabel->"In[23]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \(FindRoot::"cvnwt" \(\(:\)\(\ \)\) 
      "Newton's method failed to converge to the prescribed accuracy after \!\
\(15\) iterations."\)], "Message",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 0.00031747975225662815`}\)], "Output",
  CellLabel->"Out[23]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell[TextData[{
  "If you think that ",
  StyleBox["FindRoot", "MR"],
  " is getting close to a root but may need more steps to get there, increase \
the setting of the ",
  StyleBox["MaxIterations", "MR"],
  " option. "
}], "ExampleText",
  CellTags->"FindRoot"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[x\^4, {x, 0.0237573}, \n\tAccuracyGoal \[Rule] 24, 
      WorkingPrecision \[Rule] 34, \n\tMaxIterations \[Rule] 50]\)], "Input",
  CellLabel->"In[24]:= ",
  CellTags->"FindRoot"],

Cell[BoxData[
    \({x \[Rule] 
        4.247418856187920967704775348713908122988419235`34*^-7}\)], "Output",
  CellLabel->"Out[24]= ",
  CellTags->"FindRoot"]
}, Open  ]],

Cell["First", "ObjectName",
  CellTags->{"S0.1192", "First"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["First[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " gives the first element in ",
  StyleBox["expr", "TI"],
  ". "
}], "Usage",
  CellTags->{"S0.1192", "First"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["First[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " is equivalent to ",
  StyleBox["expr", "TI"],
  StyleBox["[[1]]", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.8.4",
    ButtonData:>{"1.8.4", "4.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Part",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Last",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Rest",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Take",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Select",
    ButtonStyle->"RefGuideLink"],
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You will have to evaluate all the cells in this example to \
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Cell["Fourier", "ObjectName",
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\[FilledSmallSquare] To ensure a unique inverse discrete Fourier transform, \
",
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Differentiating a complicated indefinite integral should yield the \
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Here both the trigonometric functions and the gamma function are \
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By contrast, here the trigonometric functions are left untouched. \
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Cell["Here is a pure function that squares its argument. ", "ExampleText",
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If your function needs several parameters, you specify them \
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\[FilledSmallSquare] ",
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Cell["\<\
The incomplete gamma function with an integer as the first argument \
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Cell[TextData[{
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Cell["\<\
Many functions can be expressed in terms of gamma functions. \
\>", \
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
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  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
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  StyleBox["xmin", "TI"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox["y", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["ymin", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["ymax", "TI"],
  Cell[BoxData[
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Cell[TextData[{
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\[FilledSmallSquare] The option ",
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    ButtonStyle->"MainBookLink"],
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If it is necessary to compute it repeatedly, it is useful to \
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Cell["\<\
Let's clean up by getting rid of the symbols defined in these \
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The gamma function generalizes factorials to numbers beyond the \
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Cell[TextData[{
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          Times[ -448, 
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The Gegenbauer polynomials are pairwise orthogonal with respect to \
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This verifies the defining differential equation for n = 2 and m = \
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\[FilledSmallSquare] The following options can be given: \
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\[FilledSmallSquare] ",
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\[FilledSmallSquare] The following options can be given: \
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variations, as well as the number of polynomials, the number of terms in each \
polynomial, the total degree of each polynomial, and the largest coefficient \
in each polynomial. "
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Here is a set of polynomials from a recent textbook. We do five \
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\[FilledSmallSquare] The following basic formats are supported for textual \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Table\\\\\\\\\\\
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\[FilledSmallSquare] ",
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  " objects. \n\[FilledSmallSquare] Animated graphics are imported as lists \
of ",
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  StyleBox["Graphics[", "MR"],
  StyleBox["data", "TI"],
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PPM\\\\\\\\\\\\\
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Cell["\<\
\[FilledSmallSquare] The following general options can be given: \
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Cell[TextData[StyleBox["GIF", "MB"]], "Subsubsection",
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Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
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  CellTags->{"Import", "S0.1243"}],

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Cell[TextData[StyleBox["HDF", "MB"]], "Subsubsection",
  CellTags->"Import"],

Cell["\<\
\[FilledSmallSquare] The following option can be given: \
\>", \
"Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[BoxData[
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Cell[TextData[{
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  " ",
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  StyleBox["\"DatasetSelection\"", "MR"],
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  StyleBox["->", "MR"],
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  StyleBox["string", "TI"],
  ". ",
  StyleBox["\"DatasetSelection\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["Information", "MR"],
  " will not import the datasets in the file but instead return a list of \
length equal to the number of datasets in the file and contain information \
about the format types, names, and dimensions of the datasets. \n\
\[FilledSmallSquare] Import of HDF currently only supports the \"scientific \
datasets\" SDS interface. "
}], "Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[TextData[StyleBox["JPEG", "MB"]], "Subsubsection",
  CellTags->"Import"],

Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
"Notes",
  CellTags->{"Import", "S0.1243"}],

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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
ColorReductionPalette\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
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          },
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  CellTags->"Import"],

Cell[TextData[{
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\[Hyphen]mapped image, ",
  StyleBox["Import", "MR"],
  " can color reduce or quantize the JPEG as it is being imported. By default \
no color reduction is done. \n\[FilledSmallSquare] Using a ",
  StyleBox["\"ColorReductionDither\"", "MR"],
  " method can provide better resulting images once a color map is chosen by \
dithering the resulting image. ",
  StyleBox["\"ColorReductionDither\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["\"Ordered\"", "MR"],
  " provides relatively fast and medium quality dithering, while ",
  StyleBox["\"ColorReductionDither\"", "MR"],
  " ",
  StyleBox["-> \"Floyd-Steinberg\"", "MR"],
  " provides the best quality error\[Hyphen]diffusion dithering, requiring \
more time and memory. ",
  StyleBox["\"ColorReductionDither\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["None", "MR"],
  " will perform no dithering on the imported image. \n\[FilledSmallSquare] \
",
  StyleBox["\"ColorReduction\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["\"Fixed\"", "MR"],
  " chooses evenly spaced colors from a color space, which may not pick the \
most optimal colors for the image. ",
  StyleBox["\"ColorReduction\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["\"Adaptive\"", "MR"],
  " uses a median\[Hyphen]cut algorithm to choose the optimal set of colors \
for an image. \n\[FilledSmallSquare] ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["Automatic", "MR"],
  " will choose an adaptive set of at most ",
  StyleBox["256", "MR"],
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  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["n", "TI"],
  " will choose at most ",
  StyleBox["n", "TI"],
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  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
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  Cell[BoxData[
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  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"color\\\\\\\"\\\", \\\
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  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
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  " specifies a list of color directives to use to color reduce the image. \
The values for ",
  Cell[BoxData[
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  ", ",
  StyleBox["GrayLevel", "MR"],
  ", ",
  StyleBox["Hue", "MR"],
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  StyleBox["RGBColor", "MR"],
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  StyleBox["\"ColorReductionPalette\"", "MR"],
  " must be between ",
  StyleBox["2", "MR"],
  " and ",
  StyleBox["256", "MR"],
  ". \n\[FilledSmallSquare] The following example produces an image using a \
color map of \"browser safe\" ",
  StyleBox["216", "MR"],
  " colors: ",
  StyleBox["Import[\"file.jpg\",", "MR"],
  " ",
  StyleBox["\"JPEG\",", "MR"],
  " ",
  StyleBox["ConversionOptions", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["{\"ColorReduction\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["\"Fixed\",", "MR"],
  " ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["216}]", "MR"],
  ". \n\[FilledSmallSquare] Color reduction of JPEG images only applies to \
JPEG images with three color components. If you are producing these JPEG \
images yourself, you can force the use of three color components with ",
  StyleBox["Export", "MR"],
  " and ",
  StyleBox["\"Colorspace\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["\"RGBColor\"", "MR"],
  ": ",
  StyleBox["Export[\"file.jpg\",", "MR"],
  " ",
  StyleBox["gr,", "MR"],
  " ",
  StyleBox["\"JPEG\",", "MR"],
  " ",
  StyleBox["ConversionOptions->{\"Colorspace\"", "MR"],
  " ",
  StyleBox["-> RGBColor}]", "MR"],
  ". \n\[FilledSmallSquare] The option ",
  StyleBox["\"Colorspace\"", "MR"],
  " can be explicitly set to ",
  StyleBox["RGBColor", "MR"],
  " or ",
  StyleBox["GrayLevel", "MR"],
  " to force the resulting imported image to a specific color space. "
}], "Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[TextData[StyleBox["List", "MB"]], "Subsubsection",
  CellTags->"Import"],

Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
"Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[BoxData[
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\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\[Backslash][Sterling]\\\\\\\\\\\
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\\!\\(\\*StyleBox[\\\"\\\\\\\"{\\\\\\\\\\\\\\\"c\\\\\\\\\\\\\\\",\\\\\\\"\\\",\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"DateStyle\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"DebitMark\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ListSeparators\\\
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\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \
\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\[Backslash]t\\\\\\\\\\\\\\\"}\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"NumberPoint\\\\\
\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\
\\\\\"\\\\\\\\\\\\\\\".\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", \
"\<\"decimal point character \"\>"}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Import"],

Cell[TextData[{
  "\[FilledSmallSquare] The option ",
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  " can be set to the values ",
  StyleBox["\"American\"", "MR"],
  ", ",
  StyleBox["\"European\"", "MR"],
  ", or ",
  StyleBox["\"Scientific\"", "MR"],
  ". "
}], "Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[TextData[{
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  StyleBox["PGM", "MB"],
  StyleBox[", ", "TB"],
  StyleBox["PPM", "MB"],
  StyleBox[", ", "TB"],
  StyleBox["PNM", "MB"]
}], "Subsubsection",
  CellTags->"Import"],

Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
"Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ColorReduction\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
ColorReductionDither\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Import"],

Cell[TextData[{
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  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
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  StyleBox["Raster", "MR"],
  " of graylevels is imported. \n\[FilledSmallSquare] Although the PBM \
formats are not stored as a color\[Hyphen]mapped image, ",
  StyleBox["Import", "MR"],
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  StyleBox["\"ColorReduction\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  " will perform color quantization. \n\[FilledSmallSquare] By dithering the \
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  StyleBox["\"ColorReductionDither\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
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\[FilledSmallSquare] ",
  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["Automatic", "MR"],
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  StyleBox["->", "MR"],
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  StyleBox["n", "TI"],
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  StyleBox["\"ColorReductionPalette\"", "MR"],
  " ",
  StyleBox["->", "MR"],
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  StyleBox[",", "MR"],
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  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"color\\\\\\\"\\\", \\\
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  StyleBox["GrayLevel", "MR"],
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  StyleBox["2", "MR"],
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}], "Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[TextData[StyleBox["Table", "MB"]], "Subsubsection",
  CellTags->"Import"],

Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
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  CellTags->{"Import", "S0.1243"}],

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\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\[Backslash][Yen]\\\\\\\\\\\\\\\
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\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\[Backslash][Euro]\\\\\\\\\\\\\\\
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\\!\\(\\*StyleBox[\\\"\\\\\\\"{\\\\\\\\\\\\\\\"c\\\\\\\\\\\\\\\",\\\\\\\"\\\",\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"NumberPoint\\\\\
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"\<\"decimal point character \"\>"},
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\[Backslash]n\\\\\\\\\\\\\\\"},\\\\\\\"\\\", \\\"MR\\\"]\\) \
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\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\[Backslash]t\\\\\\\\\\\\\\\"}}\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
TwoDigitYearFunction\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Import"],

Cell[TextData[{
  "\[FilledSmallSquare] The option ",
  StyleBox["\"DateStyle\"", "MR"],
  " can be set to the values ",
  StyleBox["\"American\"", "MR"],
  ", ",
  StyleBox["\"European\"", "MR"],
  ", or ",
  StyleBox["\"Scientific\"", "MR"],
  ". \n\[FilledSmallSquare] A field is considered to be a date if it consists \
of three adjacent, forward slash\[Hyphen]delimited integers of which the \
month, day, and year values appear to be reasonable. Months must be between 1 \
and 12 inclusive. The day must not lie outside the valid range of days for \
the given month. A year's validity can be determined indirectly by ",
  StyleBox["\"TwoDigitYearFunction\"", "MR"],
  ". \n\[FilledSmallSquare] A date field will be broken into a list of the \
form ",
  StyleBox["year", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["month", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["day", "TI"],
  ". \n\[FilledSmallSquare] The option ",
  StyleBox["\"TwoDigitYearFunction\"", "MR"],
  " can be set to ",
  StyleBox["None", "MR"],
  ", ",
  StyleBox["Automatic", "MR"],
  ", or an arbitrary function. \n\[FilledSmallSquare] Setting the ",
  StyleBox["\"TwoDigitYearFunction\"", "MR"],
  " to its default value of ",
  StyleBox["None", "MR"],
  " keeps the date field from being interpreted as a date if the year has \
only two\[Hyphen]digits. \n\[FilledSmallSquare] When ",
  StyleBox["\"TwoDigitYearFunction\"", "MR"],
  " is set to ",
  StyleBox["Automatic", "MR"],
  ", 1900 is added to the year and the field is interpreted as a date. \n\
\[FilledSmallSquare] User\[Hyphen]defined functions for ",
  StyleBox["\"TwoDigitYearFunction\"", "MR"],
  " must return ",
  StyleBox["Integer", "MR"],
  " values. "
}], "Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[TextData[StyleBox["TIFF", "MB"]], "Subsubsection",
  CellTags->"Import"],

Cell["\<\
\[FilledSmallSquare] The following options can be given: \
\>", \
"Notes",
  CellTags->{"Import", "S0.1243"}],

Cell[BoxData[
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\\\"True\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"checks image color depth \
\"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ImageIndex\\\\\\\
\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"Import"],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["\"CheckDepth\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["True", "MR"],
  " will check the data on importing to see if the actual bit depth of the \
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  StyleBox["Raster", "MR"],
  " with integer values scaled to a more exact representation of what was \
stored in the TIFF. For example, a 4\[Hyphen]bit TIFF image would be imported \
as a ",
  StyleBox["Raster", "MR"],
  " with a matrix of integer values between ",
  StyleBox["0", "MR"],
  " and ",
  StyleBox["15", "MR"],
  " and not the normal ",
  StyleBox["0", "MR"],
  " to ",
  StyleBox["255", "MR"],
  " associated with a byte of image sample information. \n\
\[FilledSmallSquare] If the TIFF contains multiple images, ",
  StyleBox["\"ImageIndex\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["i", "TI"],
  " will select a particular image to import. ",
  StyleBox["\"ImageIndex\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"i\\\\\\\"\\\", \\\"TI\
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  StyleBox[", ", "MR"],
  Cell[BoxData[
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  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  " will select a list of images from the TIFF. By default all images of a \
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}], "Notes",
  CellTags->{"Import", "S0.1243"}],

Cell["ImportString", "ObjectName",
  CellTags->{"S0.1243", "ImportString"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["ImportString[\"", "MR"],
  StyleBox["data", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["format", "TI"],
  StyleBox["\"]", "MR"],
  " imports data in the specified format from a string. "
}], "Usage",
  CellTags->{"S0.1243", "ImportString"}],

Cell[TextData[{
  "\[FilledSmallSquare] See notes for ",
  StyleBox["Import", "MR"],
  ". \n\[FilledSmallSquare] See also: ",
  ButtonBox["ExportString",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
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Cell["\<\
This variant also places copies before the first and after the last \
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Cell[BoxData[
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Cell["Install", "ObjectName",
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Cell[TextData[{
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",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["This generates a random integer in the range from 0 to 10. ", \
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["Indefinite integrals", "ExampleSubsection",
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Cell["\<\
Here are three indefinite integrals that are evaluated by the \
(extended) Risch algorithm. \
\>", "ExampleText",
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Here are two indefinite integrals that are evaluated by special \
table lookup rules. \
\>", "ExampleText",
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
  "When an integrand depends on a parameter, the indefinite integral should \
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\[OpenCurlyDoubleQuote]generic\[CloseCurlyDoubleQuote] values of the \
parameter. For certain values the reported integral may be meaningless, as is \
the case here for ",
  Cell[BoxData[
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  ". "
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  CellTags->"Integrate"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Integrate"],

Cell[BoxData[
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  CellTags->"Integrate"]
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  CellTags->"Integrate"]
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  " lets you explicitly state conditions on the parameters. If you are only \
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Cell[BoxData[
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  CellTags->"Integrate"],

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Cell[BoxData[
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  CellTags->"Integrate"],

Cell[BoxData[
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  CellLabel->"Out[10]= ",
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Integrate"],

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  CellLabel->"Out[11]= ",
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Cell[TextData[{
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  Cell[BoxData[
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  CellTags->"Integrate"],

Cell[CellGroupData[{

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  CellTags->"Integrate"]
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Cell["Here is the result when you ask for a definite integral. ", \
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  CellLabel->"In[14]:= ",
  CellTags->"Integrate"],

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Cell[TextData[{
  "Even for a definite integral, you should use results with caution if they \
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  Cell[BoxData[
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  " (that is, if ",
  Cell[BoxData[
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  Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[15]:= ",
  CellTags->"Integrate"],

Cell[BoxData[
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  CellLabel->"Out[15]= ",
  CellTags->"Integrate"]
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[16]:= ",
  CellTags->"Integrate"],

Cell[BoxData[
    \(Integrate::"idiv" \(\(:\)\(\ \)\) 
      "Integral of \!\(Tan[\(\(\[Pi]\\ x\)\/2\)]\) does not converge on \
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  CellTags->"Integrate"],

Cell[BoxData[
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  CellTags->"Integrate"]
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Cell[CellGroupData[{

Cell[TextData[{
  "Options: ",
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  CellTags->"Integrate"],

Cell[TextData[{
  "Sometimes an integrand has a singularity on the path of integration that \
causes the integral (in the Riemann sense of an area) to diverge. This is the \
case for this function, at the point ",
  Cell[BoxData[
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  ". "
}], "ExampleText",
  CellTags->"Integrate"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Integrate[\@x\/\(x - 1\), {x, 0, 3}]\)], "Input",
  CellLabel->"In[17]:= ",
  CellTags->"Integrate"],

Cell[BoxData[
    \(\(-2\) + 2\ \@3 - 2\ \((\(-1\) + Log[2])\) + 2\ Log[2] + 
      Log[\(-1\) + \@3] - Log[1 + \@3]\)], "Output",
  CellLabel->"Out[17]= ",
  CellTags->"Integrate"]
}, Open  ]],

Cell["\<\
The integral may nonetheless exist in the sense of the Cauchy \
principal value. This value is defined by considering the integral obtained \
by omitting a small interval centered at the singularity, and taking the \
limit as the interval's length shrinks to zero. \
\>", "ExampleText",
  CellTags->"Integrate"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[18]:= ",
  CellTags->"Integrate"],

Cell[BoxData[
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  CellLabel->"Out[18]= ",
  CellTags->"Integrate"]
}, Open  ]],

Cell[TextData[{
  "Cauchy principal values can also be obtained numerically, using the \
package ",
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  ". "
}], "ExampleText",
  CellTags->"Integrate"],

Cell[BoxData[
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  CellLabel->"In[19]:= ",
  CellTags->"Integrate"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[20]:= ",
  CellTags->"Integrate"],

Cell[BoxData[
    \(True\)], "Output",
  CellLabel->"Out[20]= ",
  CellTags->"Integrate"]
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Cell[BoxData[
    \(Clear[int]\)], "Input",
  CellLabel->"In[21]:= ",
  CellTags->"Integrate"]
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Cell[CellGroupData[{

Cell["Numerical integration", "ExampleSubsection",
  CellTags->"Integrate"],

Cell[TextData[{
  "When a function cannot be integrated symbolically, you can usually obtain \
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  StyleBox["NIntegrate", "MR"],
  ". "
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  CellTags->"Integrate"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Integral]\_0\%1 Sin[Sin[x]] \[DifferentialD]x\)], "Input",
  CellLabel->"In[22]:= ",
  CellTags->"Integrate"],

Cell[BoxData[
    \(\[Integral]\_0\%1 Sin[Sin[x]] \[DifferentialD]x\)], "Output",
  CellLabel->"Out[22]= ",
  CellTags->"Integrate"]
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Cell[CellGroupData[{

Cell[BoxData[
    \(NIntegrate[Sin[Sin[x]], {x, 0, 1}]\)], "Input",
  CellLabel->"In[23]:= ",
  CellTags->"Integrate"],

Cell[BoxData[
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  CellLabel->"Out[23]= ",
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Cell[TextData[{
  " See the Further Examples for ",
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Cell["InterpolatingFunction", "ObjectName",
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Cell[TextData[{
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  StyleBox["domain", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["table", "TI"],
  StyleBox["]", "MR"],
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interpolation. "
}], "Usage",
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Cell[TextData[{
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  StyleBox["InterpolatingFunction", "MR"],
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  " specifies the domain of the data from which the ",
  StyleBox["InterpolatingFunction", "MR"],
  " was constructed. \n\[FilledSmallSquare] If you supply arguments outside \
of the domain, a warning is generated, and then an extrapolated value is \
returned. \n\[FilledSmallSquare] ",
  StyleBox["InterpolatingFunction", "MR"],
  " objects that take any number of real arguments may be constructed. \n\
\[FilledSmallSquare] You can take derivatives of ",
  StyleBox["InterpolatingFunction", "MR"],
  " objects using ",
  StyleBox["D", "MR"],
  " and ",
  StyleBox["Derivative", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["NDSolve", "MR"],
  " returns its results in terms of ",
  StyleBox["InterpolatingFunction", "MR"],
  " objects. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.6.4",
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    ButtonStyle->"MainBookLink"],
  ", ",
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    ButtonStyle->"MainBookLink"],
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  ButtonBox["CompiledFunction",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FunctionInterpolation",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
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    ButtonData:>{"NumericalMath`SplineFit`"},
    ButtonStyle->"AddOnsLink"],
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}], "Notes",
  CellTags->{"S0.1256", "InterpolatingFunction"}]
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"InterpolatingFunction"],

Cell[TextData[{
  "Here is a table of values of the reciprocal function and an ",
  StyleBox["InterpolatingFunction", "MR"],
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}], "ExampleText",
  CellTags->"InterpolatingFunction"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[{x, 1\/x}, {x, 1, 10}]\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"InterpolatingFunction"],

Cell[BoxData[
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  CellLabel->"Out[1]= ",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[2]:= ",
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Cell[BoxData[
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Cell[TextData[{
  "An ",
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data points and it gives approximate values in between. "
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Cell[CellGroupData[{

Cell[BoxData[
    \({rec[5. ], rec[7.5], 1\/7.5}\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"InterpolatingFunction"],

Cell[BoxData[
    \({0.2`, 0.13330853174603174`, 0.13333333333333333`}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"InterpolatingFunction"]
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You can take the union and intersection of the two or more \
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You can do interval arithmetic with many functions. For example, \
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With ordinary machine\[Hyphen]precision arithmetic, this gives an \
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Cell[TextData[{
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Cell[TextData[{
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Cell[TextData[{
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Here is the least common multiple of the first 1000 integers. \
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Here are two cases where it helps to see the internal \
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  StyleBox["n", "TI"],
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  StyleBox["m", "TI"],
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  StyleBox["z", "TI"],
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Cell["This is the derivative. ", "ExampleText",
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\[FilledSmallSquare] See ",
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Cell[TextData[{
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This gives the first associated Legendre function of the second \
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\>", "ExampleText",
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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  " times the height of its contents. \n\[FilledSmallSquare] ",
  StyleBox["LineSpacing->", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["0,", "MR"],
  " ",
  StyleBox["n", "TI"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  " makes the total height of each line exactly ",
  StyleBox["n", "TI"],
  " printer's points. \n\[FilledSmallSquare] ",
  StyleBox["LineSpacing->", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["c", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["n", "TI"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  " makes the total height ",
  StyleBox["c", "TI"],
  " times the height of the contents plus ",
  StyleBox["n", "TI"],
  " printer's points. \n\[FilledSmallSquare] A typical default setting is ",
  StyleBox["LineSpacing->{1,", "MR"],
  " ",
  StyleBox["1}", "MR"],
  ", which leaves space for the contents of the line, plus 1 printer's point \
(approximately ",
  Cell[BoxData[
      \(TraditionalForm\`1\/72\)], "InlineFormula"],
  " of an inch) of extra space. \n\[FilledSmallSquare] ",
  StyleBox["LineSpacing->{2,", "MR"],
  " ",
  StyleBox["0}", "MR"],
  " makes text \[OpenCurlyDoubleQuote]double spaced\[CloseCurlyDoubleQuote]. \
\n\[FilledSmallSquare] ",
  StyleBox["LineSpacing->", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["1,", "MR"],
  " ",
  StyleBox["-", "MR"],
  StyleBox["n", "TI"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  " tightens text by ",
  StyleBox["n", "TI"],
  " printer's points. \n\[FilledSmallSquare] ",
  StyleBox["LineSpacing", "MR"],
  " applies both to ordinary text and ",
  StyleBox["Mathematica", "TI"],
  " expressions. \n\[FilledSmallSquare] In ordinary text, ",
  StyleBox["LineSpacing", "MR"],
  " determines the spacing between lines produced by automatic linebreaking. \
For lines produced by explicit Return characters ",
  StyleBox["ParagraphSpacing", "MR"],
  " is added. \n\[FilledSmallSquare] In ",
  StyleBox["Mathematica", "TI"],
  " expressions, ",
  StyleBox["LineSpacing", "MR"],
  " is used whether lines are produced by automatic linebreaking or by \
explicit Return characters. \n\[FilledSmallSquare] Extra space specified by \
",
  StyleBox["LineSpacing", "MR"],
  " is inserted equally above and below a line, except that no extra space is \
inserted before the first line or after the last line of an expression or \
cell. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.10",
    ButtonData:>{"2.10.10", "10.12"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["FontSize",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ParagraphSpacing",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LineSpacing"}],

Cell["LinkClose", "ObjectName",
  CellTags->{"S0.1278", "LinkClose"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkClose[", "MR"],
  StyleBox["link", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["MathLink", "TI"],
  " connection. "
}], "Usage",
  CellTags->{"S0.1278", "LinkClose"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
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  StyleBox["LinkObject", "MR"],
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  StyleBox["LinkLaunch", "MR"],
  " and ",
  StyleBox["Links", "MR"],
  ". \n\[FilledSmallSquare] Closing a ",
  StyleBox["MathLink", "TI"],
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the connection. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.6",
    ButtonData:>{"2.12.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["LinkInterrupt",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Close",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LinkClose"}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"LinkClose"],

Cell[CellGroupData[{

Cell[TextData[{
  "See the Further Examples for ",
  ButtonBox["LinkLaunch",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "ExampleSubsection",
  CellTags->"LinkClose"],

Cell["LinkConnect", "ObjectName",
  CellTags->{"S0.1278", "LinkConnect"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkConnect[\"", "MR"],
  StyleBox["name", "TI"],
  StyleBox["\"]", "MR"],
  " connects to a ",
  StyleBox["MathLink", "TI"],
  " link created by another program. "
}], "Usage",
  CellTags->{"S0.1278", "LinkConnect"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkConnect", "MR"],
  " by default operates with internet TCP links, with names of the form ",
  StyleBox["port", "TI"],
  StyleBox["@", "MR"],
  StyleBox["host", "TI"],
  ". Ports are typically specified by numbers. \n\[FilledSmallSquare] ",
  StyleBox["LinkConnect", "MR"],
  " can connect to a port on a remote computer system. \n\[FilledSmallSquare] \
On some computer systems, ",
  StyleBox["LinkConnect[", "MR"],
  " ",
  StyleBox["]", "MR"],
  " will bring up a port browser. \n\[FilledSmallSquare] ",
  StyleBox["LinkConnect", "MR"],
  " returns a ",
  StyleBox["LinkObject", "MR"],
  ". \n\[FilledSmallSquare] You can use ",
  StyleBox["LinkConnect", "MR"],
  " with ",
  StyleBox["LinkCreate", "MR"],
  " to set up peer\[Hyphen]to\[Hyphen]peer communication between two ",
  StyleBox["Mathematica", "TI"],
  " processes. \n\[FilledSmallSquare] ",
  StyleBox["LinkConnect", "MR"],
  " can be used to connect to a link created by calling ",
  StyleBox["LinkCreate", "MR"],
  " in another ",
  StyleBox["Mathematica", "TI"],
  " process. \n\[FilledSmallSquare] ",
  StyleBox["LinkConnect", "MR"],
  " can be used to connect to an external program that has created a ",
  StyleBox["MathLink", "TI"],
  " link by calling the appropriate ",
  StyleBox["MathLink", "TI"],
  " library functions. \n\[FilledSmallSquare] External programs built from ",
  StyleBox["MathLink", "TI"],
  " templates using ",
  StyleBox["mcc", "MR"],
  " and ",
  StyleBox["mprep", "MR"],
  " can typically create ",
  StyleBox["MathLink", "TI"],
  " links whenever they are given ",
  StyleBox["-linkcreate", "MR"],
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  StyleBox["LinkProtocol", "MR"],
  " specifies the underlying data transport protocol that ",
  StyleBox["LinkConnect", "MR"],
  " should use. \n\[FilledSmallSquare] ",
  StyleBox["LinkConnect", "MR"],
  " internally calls a function analogous to the ",
  StyleBox["MLOpenArgv()", "MR"],
  " function in the ",
  StyleBox["MathLink", "TI"],
  " library. \n\[FilledSmallSquare] Even though no program may yet be \
connected to the other end of the ",
  StyleBox["MathLink", "TI"],
  " link, the function ",
  StyleBox["LinkConnect", "MR"],
  " will return immediately and will not block. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.6",
    ButtonData:>{"2.12.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["LinkCreate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkLaunch",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkClose",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LinkConnect"}],

Cell["LinkCreate", "ObjectName",
  CellTags->{"S0.1278", "LinkCreate"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkCreate[\"", "MR"],
  StyleBox["name", "TI"],
  StyleBox["\"]", "MR"],
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  StyleBox["MathLink", "TI"],
  " link with the specified name for another program to connect to. \n\
\[FilledSmallSquare] ",
  StyleBox["LinkCreate[", "MR"],
  " ",
  StyleBox["]", "MR"],
  " picks an unused port on your computer system and creates a ",
  StyleBox["MathLink", "TI"],
  " link on it. "
}], "Usage",
  CellTags->{"S0.1278", "LinkCreate"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkCreate", "MR"],
  " returns a ",
  StyleBox["LinkObject", "MR"],
  ". \n\[FilledSmallSquare] You can use ",
  StyleBox["LinkCreate", "MR"],
  " and ",
  StyleBox["LinkConnect", "MR"],
  " to set up peer\[Hyphen]to\[Hyphen]peer communication between two ",
  StyleBox["Mathematica", "TI"],
  " processes. \n\[FilledSmallSquare] The option ",
  StyleBox["LinkProtocol", "MR"],
  " specifies the underlying data transport protocol to use. \n\
\[FilledSmallSquare] ",
  StyleBox["LinkCreate", "MR"],
  " internally calls a function analogous to the ",
  StyleBox["MLOpenArgv()", "MR"],
  " function in the ",
  StyleBox["MathLink", "TI"],
  " library. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.6",
    ButtonData:>{"2.12.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["LinkConnect",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkLaunch",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkClose",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LinkCreate"}],

Cell["LinkInterrupt", "ObjectName",
  CellTags->{"S0.1278", "LinkInterrupt"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkInterrupt[", "MR"],
  StyleBox["link", "TI"],
  StyleBox["]", "MR"],
  " sends an interrupt to the program at the other end of the specified ",
  StyleBox["MathLink", "TI"],
  " connection. "
}], "Usage",
  CellTags->{"S0.1278", "LinkInterrupt"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
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  StyleBox["LinkObject", "MR"],
  ", as returned by functions such as ",
  StyleBox["LinkLaunch", "MR"],
  " or ",
  StyleBox["Links", "MR"],
  ". \n\[FilledSmallSquare] It is up to the external program to determine how \
it will handle the interrupt. \n\[FilledSmallSquare] External programs \
created from ",
  StyleBox["MathLink", "TI"],
  " templates will by default set the global variable ",
  StyleBox["MLAbort", "MR"],
  " if they receive an abort. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.7",
    ButtonData:>{"2.12.7", "7.33"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Interrupt",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkClose",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LinkInterrupt"}],

Cell["LinkLaunch", "ObjectName",
  CellTags->{"S0.1278", "LinkLaunch"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkLaunch[\"", "MR"],
  StyleBox["prog", "TI"],
  StyleBox["\"]", "MR"],
  " starts the external program ",
  StyleBox["prog", "TI"],
  " and opens a ",
  StyleBox["MathLink", "TI"],
  " connection to it. "
}], "Usage",
  CellTags->{"S0.1278", "LinkLaunch"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkLaunch[\"", "MR"],
  StyleBox["prog", "TI"],
  StyleBox["\"]", "MR"],
  " runs ",
  StyleBox["prog", "TI"],
  " as a subsidiary or child process to your current ",
  StyleBox["Mathematica", "TI"],
  " session. \n\[FilledSmallSquare] You can use a command such as ",
  StyleBox["LinkLaunch[\"math", "MR"],
  " ",
  StyleBox["-mathlink\"]", "MR"],
  " to launch a subsidiary ",
  StyleBox["Mathematica", "TI"],
  " kernel process from within your ",
  StyleBox["Mathematica", "TI"],
  " session. \n\[FilledSmallSquare] On most computer systems calling ",
  StyleBox["LinkLaunch[\"", "MR"],
  StyleBox["prog", "TI"],
  StyleBox["\"]", "MR"],
  " multiple times with the same argument will start several ",
  StyleBox["prog", "TI"],
  " processes running. On systems such as the Macintosh, however, only one \
program with any particular name can be running at the same time. \n\
\[FilledSmallSquare] On some computer systems, ",
  StyleBox["LinkLaunch[", "MR"],
  " ",
  StyleBox["]", "MR"],
  " will bring up a program browser. \n\[FilledSmallSquare] ",
  StyleBox["LinkLaunch", "MR"],
  " returns a ",
  StyleBox["LinkObject", "MR"],
  ". \n\[FilledSmallSquare] The option ",
  StyleBox["LinkProtocol", "MR"],
  " specifies the underlying data transport protocol to use. \n\
\[FilledSmallSquare] ",
  StyleBox["LinkLaunch", "MR"],
  " internally calls a function analogous to the ",
  StyleBox["MLOpenArgv()", "MR"],
  " function in the ",
  StyleBox["MathLink", "TI"],
  " library. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.7",
    ButtonData:>{"2.12.7", "7.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Install",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkCreate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkConnect",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkClose",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LinkLaunch"}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"LinkLaunch"],

Cell[TextData[{
  StyleBox["LinkLaunch", "MR"],
  " launches and connects to a program that is aware of ",
  StyleBox["MathLink", "TI"],
  ". Here we launch a copy of ",
  StyleBox["Mathematica", "TI"],
  "; a reason for doing this would be for two or more kernels to cooperate in \
solving a problem. This command works on any platform except the Macintosh. "
}], "ExampleText",
  CellTags->"LinkLaunch"],

Cell[CellGroupData[{

Cell[BoxData[
    \(link = LinkLaunch["\<math -mathlink\>"]\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"LinkLaunch"],

Cell[BoxData[
    \(LinkObject["math -mathlink", 14, 3]\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"LinkLaunch"]
}, Open  ]],

Cell[TextData[{
  " On the Macintosh you need to make a duplicate of MathKernel because you \
cannot launch an application from a single file more than once. Single quotes \
must surround the pathname if it contains spaces. This would be the command \
on the Macintosh: ",
  StyleBox["link", "MR"],
  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["LinkLaunch[\"'MathKernel", "MR"],
  " ",
  StyleBox["copy'\"]", "MR"],
  " "
}], "ExampleText",
  CellTags->"LinkLaunch"],

Cell[TextData[{
  StyleBox["MathLink", "TI"],
  " communicates using packets. When the second kernel starts up, it sends a \
packet containing the first input prompt to the program that launched it. "
}], "ExampleText",
  CellTags->"LinkLaunch"],

Cell[CellGroupData[{

Cell[BoxData[
    \(LinkRead[link]\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"LinkLaunch"],

Cell[BoxData[
    \(InputNamePacket["In[1]:= "]\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"LinkLaunch"]
}, Open  ]],

Cell[TextData[{
  "To send an expression to the second kernel for evaluation, wrap it in an \
EvaluatePacket; the result comes back in a ",
  StyleBox["ReturnPacket", "MR"],
  ". The ",
  StyleBox["Unevaluated", "MR"],
  " around the expression keeps it from evaluating before it is sent. "
}], "ExampleText",
  CellTags->"LinkLaunch"],

Cell[CellGroupData[{

Cell[BoxData[
    \({LinkWrite[link, Unevaluated[EvaluatePacket[2 + 2]]], 
      LinkRead[link]}\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"LinkLaunch"],

Cell[BoxData[
    \({Null, ReturnPacket[4]}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"LinkLaunch"]
}, Open  ]],

Cell["When you are finished, quit the second kernel. ", "ExampleText",
  CellTags->"LinkLaunch"],

Cell[BoxData[
    \(LinkClose[link]\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"LinkLaunch"],

Cell["LinkObject", "ObjectName",
  CellTags->{"S0.1278", "LinkObject"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkObject[\"", "MR"],
  StyleBox["name", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
  " is an object that represents an active ",
  StyleBox["MathLink", "TI"],
  " connection for functions such as ",
  StyleBox["LinkRead", "MR"],
  " and ",
  StyleBox["LinkWrite", "MR"],
  ". "
}], "Usage",
  CellTags->{"S0.1278", "LinkObject"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkConnect", "MR"],
  ", ",
  StyleBox["LinkCreate", "MR"],
  ", ",
  StyleBox["LinkLaunch", "MR"],
  " and ",
  StyleBox["Install", "MR"],
  " all return ",
  StyleBox["LinkObject", "MR"],
  " objects. \n\[FilledSmallSquare] The integer ",
  StyleBox["n", "TI"],
  " is a serial number used to distinguish links with the same name. \n\
\[FilledSmallSquare] ",
  StyleBox["name", "TI"],
  " is typically the name of an external program, or an internet TCP \
specification of the form ",
  StyleBox["port", "TI"],
  StyleBox["@", "MR"],
  StyleBox["host", "TI"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.2",
    ButtonData:>{"2.12.2", "2.4"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.12.9",
    ButtonData:>{"2.12.9", "9.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Links",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkReadyQ",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["InputStream",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LinkObject"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"LinkObject"],

Cell[CellGroupData[{

Cell[TextData[{
  "See the Further Examples for ",
  ButtonBox["LinkLaunch",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "ExampleSubsection",
  CellTags->"LinkObject"],

Cell["LinkPatterns", "ObjectName",
  CellTags->{"S0.1278", "LinkPatterns"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkPatterns[", "MR"],
  StyleBox["link", "TI"],
  StyleBox["]", "MR"],
  " gives a list of the patterns for which definitions were set up when the \
external program associated with the specified ",
  StyleBox["MathLink", "TI"],
  " connection was installed. "
}], "Usage",
  CellTags->{"S0.1278", "LinkPatterns"}],

Cell[TextData[{
  "\[FilledSmallSquare] Each element of the list returned by ",
  StyleBox["LinkPatterns", "MR"],
  " is wrapped in ",
  StyleBox["HoldForm", "MR"],
  " to prevent evaluation. \n\[FilledSmallSquare] The patterns in ",
  StyleBox["LinkPatterns", "MR"],
  " typically originate in ",
  StyleBox[":Pattern:", "MR"],
  " specifications in the ",
  StyleBox["MathLink", "TI"],
  " templates used to create source code for the external program. \n\
\[FilledSmallSquare] ",
  StyleBox["Uninstall[", "MR"],
  StyleBox["link", "TI"],
  StyleBox["]", "MR"],
  " calls ",
  StyleBox["Unset", "MR"],
  " on the patterns in ",
  StyleBox["LinkPatterns[", "MR"],
  StyleBox["link", "TI"],
  StyleBox["]", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.3",
    ButtonData:>{"2.12.3", "3.27"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Install",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1278", "LinkPatterns"}],

Cell["LinkProtocol", "ObjectName",
  CellTags->{"S0.1278", "LinkProtocol"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkProtocol", "MR"],
  " is an option to ",
  StyleBox["LinkLaunch", "MR"],
  ", ",
  StyleBox["Install", "MR"],
  " and related functions which specifies the underlying data transport \
protocol to use for a new ",
  StyleBox["MathLink", "TI"],
  " link. "
}], "Usage",
  CellTags->{"S0.1278", "LinkProtocol"}],

Cell[TextData[{
  "\[FilledSmallSquare] Typical settings for ",
  StyleBox["LinkProtocol", "MR"],
  " are ",
  StyleBox["\"TCP\"", "MR"],
  " (Unix), ",
  StyleBox["\"FM\"", "MR"],
  " (Microsoft Windows), ",
  StyleBox["\"PPC\"", "MR"],
  " (Macintosh) and ",
  StyleBox["\"Pipes\"", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.5",
    ButtonData:>{"2.12.5", "5.3"},
    ButtonStyle->"MainBookLink"],
  "."
}], "Notes",
  CellTags->{"S0.1278", "LinkProtocol"}],

Cell["LinkRead", "ObjectName",
  CellTags->{"S0.1278", "LinkRead"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["LinkRead[", "MR"],
  StyleBox["link", "TI"],
  StyleBox["]", "MR"],
  " reads one expression from the specified ",
  StyleBox["MathLink", "TI"],
  " connection. \n\[FilledSmallSquare] ",
  StyleBox["LinkRead[", "MR"],
  StyleBox["link", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["h", "TI"],
  StyleBox["]", "MR"],
  " wraps ",
  StyleBox["h", "TI"],
  " around the expression read before evaluating it. "
}], "Usage",
  CellTags->{"S0.1278", "LinkRead"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["link", "TI"],
  " must be an active ",
  StyleBox["LinkObject", "MR"],
  ", as returned by functions like ",
  StyleBox["LinkLaunch", "MR"],
  " or ",
  StyleBox["Links", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["LinkRead", "MR"],
  " will wait until it has read a complete expression before returning. \n\
\[FilledSmallSquare] You can test whether an expression is ready to be read \
from a particular link using ",
  StyleBox["LinkReadyQ", "MR"],
  ". \n\[FilledSmallSquare] You can use ",
  StyleBox["LinkRead[", "MR"],
  StyleBox["link", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["Hold]", "MR"],
  " to get an expression from a link without evaluating it. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.6",
    ButtonData:>{"2.12.6", "6.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["LinkReadyQ",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinkWrite",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Read",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
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Matrices such as this one can be used to illustrate how \
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            {" ", "X", "Y", "Z", " ", " ", " "},
            {" ", " ", "X", "Y", "Z", " ", " "},
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  StyleBox["6", "MR"],
  " ",
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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  ", ",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["This produces a fractal. ", "ExampleText",
  CellTags->"ListCorrelate"],

Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell[BoxData[
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  CellTags->"ListCorrelate"],

Cell["ListDensityPlot", "ObjectName",
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Cell[TextData[{
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  StyleBox["]", "MR"],
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Cell[TextData[{
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  StyleBox["ListDensityPlot", "MR"],
  " has the same options as ",
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  ". \n\[FilledSmallSquare] Successive rows of ",
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  " are arranged up the page; successive columns across the page. \n\
\[FilledSmallSquare] See notes for ",
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    ButtonStyle->"MainBookLink"],
  "."
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"ListDensityPlot"],

Cell["Here is a density plot made from a table of data. ", "ExampleText",
  CellTags->"ListDensityPlot"],

Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell[BoxData[
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    data =  . \)], "Input",
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  CellTags->"ListDensityPlot"],

Cell["ListInterpolation", "ObjectName",
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Cell[TextData[{
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  " assumes grid lines at integer positions in each direction. \n\
\[FilledSmallSquare] ",
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  StyleBox["domain", "TI"],
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Cell["Further Examples", "ExampleSection",
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  " is similar to Interpolation, but provides a more convenient interface for \
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Cell["\<\
Here is a table of values of a function on a regular three \
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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  CellTags->"ListInterpolation"]
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Cell["\<\
The approximation reproduces the values at each of the points in \
the table. \
\>", "ExampleText",
  CellTags->"ListInterpolation"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[3]:= ",
  CellTags->"ListInterpolation"],

Cell[BoxData[
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  CellTags->"ListInterpolation"]
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Cell["\<\
You can get approximate values at other points. In this case, the \
interpolation is a fairly good approximation to the function. \
\>", \
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  CellTags->"ListInterpolation"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[4]:= ",
  CellTags->"ListInterpolation"],

Cell[BoxData[
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  CellTags->"ListInterpolation"]
}, Open  ]],

Cell[TextData[{
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  ", ",
  Cell[BoxData[
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  " and ",
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  " is reduced automatically. "
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  CellTags->"ListInterpolation"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"ListInterpolation"],

Cell[BoxData[
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  CellTags->"ListInterpolation"],

Cell[BoxData[
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  CellTags->"ListInterpolation"]
}, Open  ]],

Cell["\<\
Again, the given values are represented by the approximate \
function. \
\>", "ExampleText",
  CellTags->"ListInterpolation"],

Cell[CellGroupData[{

Cell[BoxData[
    \(approxD[1, 1]\)], "Input",
  CellLabel->"In[6]:= ",
  CellTags->"ListInterpolation"],

Cell[BoxData[
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  CellLabel->"Out[6]= ",
  CellTags->"ListInterpolation"]
}, Open  ]],

Cell["The given derivatives are also represented. ", "ExampleText",
  CellTags->"ListInterpolation"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"ListInterpolation"],

Cell[BoxData[
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  CellLabel->"Out[7]= ",
  CellTags->"ListInterpolation"]
}, Open  ]],

Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"ListInterpolation"],

Cell[BoxData[
    \(\(-2\)\ c + 2\ d - dy\)], "Output",
  CellLabel->"Out[8]= ",
  CellTags->"ListInterpolation"]
}, Open  ]],

Cell["You can also get approximate values at other points. ", "ExampleText",
  CellTags->"ListInterpolation"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[9]:= ",
  CellTags->"ListInterpolation"],

Cell[BoxData[
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  CellTags->"ListInterpolation"]
}, Open  ]],

Cell["\<\
Let's clean up by getting rid of the symbols defined in these \
examples. \
\>", "ExampleText",
  CellTags->"ListInterpolation"],

Cell[BoxData[
    \(Clear[data, approx, approxD]\)], "Input",
  CellLabel->"In[10]:= ",
  CellTags->"ListInterpolation"],

Cell["ListPlay", "ObjectName",
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Cell[TextData[{
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  Cell[BoxData[
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  " \[Ellipsis] ",
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  StyleBox["]", "MR"],
  " plays a sound whose amplitude is given by the sequence of levels ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"a\\\\\\\"\\\", \\\"TI\
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Cell[TextData[{
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  StyleBox["Sound", "MR"],
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}], "Notes",
  CellTags->{"S0.1283", "ListPlay"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"DisplayFunction\\\\\\\"\\\", \\\
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\\\"MR\\\"]\\) \"\>", "\<\"function for generating output \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Epilog\\\\\\\"\\\", \\\"MR\\\"]\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"PlayRange\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Automatic\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", \("the range of amplitude levels to include "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Prolog\\\\\\\"\\\", \\\"MR\\\"]\
\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"{}\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", \("sound or graphics to be used as a prolog "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"SampleDepth\\\\\\\"\\\", \\\"MR\
\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"8\\\\\\\"\\\", \\\"MR\\\"]\
\\) \"\>", \("how many bits to use to represent each amplitude level "\)},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"SampleRate\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"8192\\\\\\\"\\\", \\\
\"MR\\\"]\\) \"\>", \("how many times per second amplitude samples should be \
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          },
        ColumnAlignments->{Left}], TraditionalForm]], "3ColumnTable",
  CellTags->"ListPlay"],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["ListPlay[", "MR"],
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",
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We compute derivatives, integrals and series expansions where m = \
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  StyleBox["m", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["-", "MR"],
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  StyleBox["n", "TI"],
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  StyleBox["Quotient[", "MR"],
  StyleBox["m", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["Mod[", "MR"],
  StyleBox["m", "TI"],
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  StyleBox["n", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["d", "TI"],
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  StyleBox["m", "TI"],
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  StyleBox["-", "MR"],
  " ",
  StyleBox["n", "TI"],
  " ",
  StyleBox["Quotient[", "MR"],
  StyleBox["m", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["n", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["d", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["Mod", "MR"],
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\[FilledSmallSquare] ",
  StyleBox["Mod[", "MR"],
  StyleBox["x", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["1]", "MR"],
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  StyleBox["x", "TI"],
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  ButtonBox["Section 1.1.3",
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    ButtonStyle->"MainBookLink"],
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  ButtonBox["Quotient",
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  ", ",
  ButtonBox["FractionalPart",
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  ", ",
  ButtonBox["MantissaExponent",
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  ", ",
  ButtonBox["PolynomialMod",
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  ButtonBox["PolynomialRemainder",
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["ModularLambda", "ObjectName",
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Cell[TextData[{
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  " and ",
  Cell[BoxData[
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  " reverses this correspondence, that is, ",
  Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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  Cell[BoxData[
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Cell[CellGroupData[{

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Cell["Module", "ObjectName",
  CellTags->{"S0.1316", "Module"}],

Cell[TextData[{
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  StyleBox["Module[", "MR"],
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  StyleBox["x", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["y", "TI"],
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  StyleBox["expr", "TI"],
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Cell[TextData[{
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module. \n\[FilledSmallSquare] ",
  StyleBox["Module", "MR"],
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it is called. \n\[FilledSmallSquare] ",
  StyleBox["Module", "MR"],
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  StyleBox["$", "MR"],
  StyleBox["nnn", "TI"],
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  StyleBox["$ModuleNumber", "MR"],
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  StyleBox["$ModuleNumber", "MR"],
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  StyleBox["Module", "MR"],
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  Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
The most common way that modules are used is to set up temporary or \
intermediate variables inside functions you define. It is important to make \
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into trouble whenever their names happen to coincide with the names of other \
variables.\
\>", "ExampleText",
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Cell[TextData[{
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Cell[BoxData[
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  CellTags->"Module"],

Cell[TextData[{
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  StyleBox["t", "MR"],
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
You can treat local variables in modules just like other symbols. \
Thus, for example, you can us them as names for local functions, you can \
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\>", "ExampleText",
  CellTags->"Module"],

Cell[TextData[{
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  StyleBox["f", "MR"],
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}], "ExampleText",
  CellTags->"Module"],

Cell[BoxData[
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  CellTags->"Module"],

Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[9]:= ",
  CellTags->"Module"],

Cell[BoxData[
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}, Open  ]],

Cell[TextData[{
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the variable in a purely symbolic way, even if there was a global value \
defined for the variable outside the module."
}], "ExampleText",
  CellTags->"Module"],

Cell[TextData[{
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  CellTags->"Module"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[10]:= ",
  CellTags->"Module"],

Cell[BoxData[
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  CellTags->"Module"]
}, Open  ]],

Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Module"],

Cell[BoxData[
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  CellLabel->"Out[11]= ",
  CellTags->"Module"]
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Cell[TextData[{
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  StyleBox["t", "MR"],
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  StyleBox["Expand", "MR"],
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}], "ExampleText",
  CellTags->"Module"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Module"],

Cell[BoxData[
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  CellTags->"Module"]
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Cell["Modulus", "ObjectName",
  CellTags->{"S0.1317", "Modulus"}],

Cell[TextData[{
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  StyleBox["Modulus->", "MR"],
  StyleBox["n", "TI"],
  " is an option that can be given in certain algebraic functions to specify \
that integers should be treated modulo ",
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  " expresses the closed\[Hyphen]form solution of this interesting \
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correct value of infinity! "
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  " once, instead of for each number used to generate the plot. \nNot too \
surprisingly, this plot looks very much like the square root function. What \
do you think would happen if you tried to solve in the other direction? "
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  " can handle higher order equations. Here is a plot of the interesting \
solution of a third\[Hyphen]order equation. \nNote that you have to specify \
enough initial conditions to uniquely determine the solution. "
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Cell["\<\
Solving systems of equations works similarly. For systems of two \
equations a so\[Hyphen]called phase plot is often a good way to visualize the \
solution. Here is a phase plot that describes the motion of a weakly damped \
pendulum. \
\>", "ExampleText",
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Cell["Ordinary differential equations: Options", "ExampleSubsection",
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Cell["\<\
This way of solving the differential equation for the cosine \
function doesn't work over such a long interval. \
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",
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Cell[TextData[{
  "The ",
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stiff is ",
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Cell["\<\
For the R\[ODoubleDot]ssler equations, the Runge\[Hyphen]Kutta \
method needs about half as many steps as the default method. \
\>", \
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Cell["Evaluate the cell to see the graphic. ", "Text",
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  " One way to get a very precise solution of an ODE is to give a \
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each arithmetic operation and function evaluation increase, but the number of \
steps typically increases exponentially also. "
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different values of ",
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Cell["Ordinary differential equations: Boundary value problems", \
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Cell[TextData[{
  "Simple linear boundary value problems can be solved by constructing the \
boundary value equations appropriately. If the order of the equation is ",
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  " you need to know the values of some combination of the function or its \
derivatives at ",
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Cell[TextData[{
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Cell["\<\
Here is a third\[Hyphen]order equation where the function values \
and combinations of the derivatives are known at three points. \
\>", \
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Cell["\<\
Not all linear equations with boundary values can be solved by the \
method that is implemented. Nonlinear equations cannot be solved, either. \
\
\>", "ExampleText",
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Cell["Evaluating this generates a lot of messages and no result. ", "Text",
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Cell["Partial differential equations: Basic usage", "ExampleSubsection",
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Cell["\<\
 For the first three examples, consider the one\[Hyphen]dimensional \
heat equation. With fixed boundary conditions this is a model for diffusion \
of heat in an insulated rod with the temperatures at the endpoints held \
fixed.\
\>", "ExampleText",
  CellTags->"NDSolve"],

Cell[TextData[{
  "This command solves the heat equation with the left end (",
  Cell[BoxData[
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  ") held at fixed temperature ",
  Cell[BoxData[
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  Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(solution = 
      NDSolve[{\[PartialD]\_t u[x, t] == \[PartialD]\_\(x, x\)u[x, t], 
          u[x, 0] == 6\ x - 6\ x\^4 + x\^2, u[0, t] == 0, u[1, t] == 1}, 
        u, {x, 0, 1}, {t, 0,  .5}]\)], "Input",
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Cell[BoxData[
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Cell[TextData[{
  " Just as for ODE's, the solution is returned as a rule to replace the \
dependent variable with an ",
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  " object. The only noticeable difference is that the ",
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  " object is two\[Hyphen]dimensional. That is to say, it takes two arguments \
to evaluate, with the arguments in the same order as the variables in the ",
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Cell[TextData[{
  "The boundary conditions can be a linear combination of Dirichlet and \
Neumann type conditions. For example, this command solves a model for the \
diffusion of heat in a one\[Hyphen]dimensional insulated rod with the left \
end held at a constant temperature and the right end radiating into free \
space. The Neumann boundary condition was entered using the ",
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  " ",
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  " ",
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  " ",
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  " ",
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Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell["\<\
An even more interesting case is when the boundary conditions are \
time\[Hyphen]dependent. For example, entering these command produces a plot \
of a solution with the temperature at the left edge varying sinusoidally. \
\
\>", "ExampleText",
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Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell["\<\
It is also possible to compute the solution of some systems of \
PDE's. These commands solve a system of mixed parabolic\[Hyphen]hyperbolic \
type and produce separate contour plots for each of the dependent variables. \
\
\>", "ExampleText",
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Cell["Evaluate the cell to see the graphics. ", "Text",
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Cell[TextData[{
  " Periodic boundary conditions are frequently convenient for numerical \
solutions. You can tell ",
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  " to solve with periodic boundary conditions by specifying that the values \
of the solution are to be equal at the left and right edges of the domain in \
one independent variable for all values of the other independent variable."
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Cell[TextData[{
  "For example, an interesting solution with periodic solutions is the \
nonlinear Schr\[ODoubleDot]dinger equation. These commands set up a periodic \
initial condition (it happens to be a soliton), and computes the solution, \
and then produces ",
  Cell[BoxData[
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  " plots showing the modulus and real and imaginary parts of the solution. "
}], "ExampleText",
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Cell["Evaluate the cell to see the graphics. ", "Text",
  CellTags->"NDSolve"],

Cell[BoxData[
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      Module[{omega, ek, xx = x - x0}, 
        omega = \((2\ m - 1)\)\ a\^2 - b\^2; \@m\ a\ JacobiCN[
            a\ \((xx - 2\ b\ t)\), 
            m]\ Exp[\[ImaginaryI]\ \((b\ xx + omega\ t)\)]]; Clear[per]; 
    per[m_] := 4\ EllipticK[m]; m = mm /. FindRoot[per[mm] == 8, {mm,  .5}]; 
    solution = 
      u /. First[
          NDSolve[{\[ImaginaryI]\ \[PartialD]\_t u[x, 
                        t] + \[PartialD]\_{x, 2}u[x, t] + 
                  2\ Abs[u[x, t]]\^2\ u[x, t] == 0, 
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              u[0, t] == u[8, t]}, u, {x, 0, 8}, {t, 0, 4}]]; 
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        PlotStyle \[Rule] {RGBColor[0, 0, 0], RGBColor[1, 0, 0], 
            RGBColor[0, 1, 0]}, PlotRange \[Rule] {\(- .8\),  .8}], {t, 0, 
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Cell[CellGroupData[{

Cell["Partial differential equations: Limitations", "ExampleSubsection",
  CellTags->"NDSolve"],

Cell[TextData[{
  StyleBox["NDSolve", "MR"],
  " uses a method for solving PDE's that is called the numerical method of \
lines. It discretizes in one variable to make a system of ODE's. This system \
is then solved using the ODE methods built into ",
  StyleBox["NDSolve", "MR"],
  "."
}], "ExampleText",
  CellTags->"NDSolve"],

Cell["\<\
For the method to work, an initial function must be specified for \
one variable and boundary values may be specified for the other variable. The \
initial function is used to find the initial conditions for the system of the \
ODE's. Boundary and initial values may be specified on at most three sides of \
a rectangle.\
\>", "ExampleText",
  CellTags->"NDSolve"],

Cell["\<\
The method has the advantage that it can solve a reasonably large \
class of equations. However, there are types of equations which it cannot \
solve.\
\>", "ExampleText",
  CellTags->"NDSolve"],

Cell["\<\
Because elliptic problems are ill\[Hyphen]posed unless boundary \
values are specified on all sides of a region, this method cannot find \
solutions of elliptic problems. A classic example is Laplace's equation. \
Entering this command indicates what will happen when you try to do something \
of this sort. Typically ill\[Hyphen]posedness will appear as numerical \
instability, and looking at the scale on the plot produced indicates that you \
should heed the message seriously! \
\>", "ExampleText",
  CellTags->"NDSolve"],

Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell["\<\
Another class of problems that the method cannot currently handle \
are those that form singularities in the solution. The discretization is done \
with finite difference methods, so fronts may be incorrectly resolved or \
completely lost. For example, Burgers' equation is a model for some of the \
features of gas dynamics, including shock formation. Entering these commands \
will produce a plot that show the oscillations typical of the interaction \
between finite difference methods and fronts. \
\>", "ExampleText",
  CellTags->"NDSolve"],

Cell["Evaluate the cell to see the graphic. ", "Text",
  CellTags->"NDSolve"],

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Cell["Partial differential equations: Options", "ExampleSubsection",
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Cell[TextData[{
  "Many of the options that control the ODE solver also apply to the PDE \
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in two stages. First the equation is discretized, and then the resulting \
system of ODE's is solved using ",
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two stages, you can specify the option value as a list. (The order of the \
independent variables in the command determines to what variable the options \
apply.)"
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Cell[TextData[{
  "In the discretization stage, the default used is fourth\[Hyphen]order \
finite differences. In some cases fourth\[Hyphen]order is not optimal; you \
can control this with the ",
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equation), it is better to increase the order."
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Cell[TextData[{
  "Entering these commands computes a solution to Airy's equation with \
periodic boundary conditions and produces ",
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",
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Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell["We clean up by clearing the definitions that were made. ", "ExampleText",
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Functions that cannot be integrated using algebraic routines can \
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Cell[BoxData[
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Cell["NonCommutativeMultiply", "ObjectName",
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Cell[TextData[{
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  StyleBox["b", "TI"],
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Cell[TextData[{
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\[FilledSmallSquare] You can use ",
  StyleBox["NonCommutativeMultiply", "MR"],
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objects. \n\[FilledSmallSquare] See ",
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Cell["NotebookClose", "ObjectName",
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Cell[TextData[{
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Cell["NotebookDelete", "ObjectName",
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Cell[TextData[{
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  ButtonBox["NotebookWrite",
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  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookDelete"}],

Cell["NotebookFind", "ObjectName",
  CellTags->{"S0.1329", "NotebookFind"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookFind[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["data", "TI"],
  StyleBox["]", "MR"],
  " sets the current selection in the specified notebook object to be the \
next occurrence of ",
  StyleBox["data", "TI"],
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  StyleBox["NotebookFind[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["data", "TI"],
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  " ",
  StyleBox["Previous]", "MR"],
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\[FilledSmallSquare] ",
  StyleBox["NotebookFind[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["data", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["All]", "MR"],
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",
  StyleBox["NotebookFind[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["data", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["dir", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["elems", "TI"],
  StyleBox["]", "MR"],
  " searches the elements of cells specified by ",
  StyleBox["elems", "TI"],
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}], "Usage",
  CellTags->{"S0.1329", "NotebookFind"}],

Cell[TextData[{
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  StyleBox["$Failed", "MR"],
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  StyleBox["data", "TI"],
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  StyleBox["notebook", "TI"],
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  StyleBox["NotebookObject", "MR"],
  ", as returned by ",
  StyleBox["NotebookOpen", "MR"],
  ", etc. \n\[FilledSmallSquare] ",
  StyleBox["data", "TI"],
  " can be a string, box expressions, or a complete cell. \n\
\[FilledSmallSquare] The possible elements are: "
}], "Notes",
  CellTags->{"S0.1329", "NotebookFind"}],

Cell[BoxData[
    FormBox[GridBox[{
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"CellContents\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"contents of each cell, represented as a string \
\"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"CellLabel\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"setting for the \
\\!\\(\\*StyleBox[\\\"\\\\\\\"CellLabel\\\\\\\"\\\", \\\"MR\\\"]\\) option of \
each cell \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"CellStyle\\\\\\\"\\\", \\\"MR\\\
\"]\\) \"\>", "\<\"name of style for each cell \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"CellTags\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"parts of the setting for the \
\\!\\(\\*StyleBox[\\\"\\\\\\\"CellTags\\\\\\\"\\\", \\\"MR\\\"]\\) option for \
each cell \"\>"},
          {
            RowBox[{
              StyleBox["{",
                "InlineFormula"], 
              StyleBox[
                
                SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"elem\\\\\\\"\\\
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                "InlineFormula"], \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", 
              StyleBox[
                
                SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"elem\\\\\\\"\\\
\", \\\"TI\\\"]\\)\"\>", "2"],
                
                "InlineFormula"], \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \[Ellipsis] \
\"\>", 
              StyleBox["}",
                
                "InlineFormula"], "\<\" \"\>"}], "\<\"list of different types \
of elements \"\>"}
          },
        ColumnAlignments->{Left}], TraditionalForm]], "2ColumnTable",
  CellTags->"NotebookFind"],

Cell[TextData[{
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  StyleBox["CellContents", "MR"],
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  StyleBox["AutoScroll->False", "MR"],
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  StyleBox["NotebookFind", "MR"],
  " is visible. \n\[FilledSmallSquare] The front end will also usually \
highlight the region corresponding to the result. \n\[FilledSmallSquare] See \
",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.28"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookLocate",
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  ", ",
  ButtonBox["SelectionMove",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookOpen",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Find",
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  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookFind"}],

Cell["NotebookGet", "ObjectName",
  CellTags->{"S0.1329", "NotebookGet"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookGet[", "MR"],
  StyleBox["obj", "TI"],
  StyleBox["]", "MR"],
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notebook object ",
  StyleBox["obj", "TI"],
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  StyleBox["NotebookGet[", "MR"],
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  StyleBox["]", "MR"],
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}], "Usage",
  CellTags->{"S0.1329", "NotebookGet"}],

Cell[TextData[{
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expression corresponding to it in the kernel. \n\[FilledSmallSquare] ",
  StyleBox["NotebookGet", "MR"],
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  StyleBox["Notebook", "MR"],
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.2",
    ButtonData:>{"2.10.2", "2.16"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookOpen",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookPut",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Get",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1329", "NotebookGet"}],

Cell["NotebookLocate", "ObjectName",
  CellTags->{"S0.1329", "NotebookLocate"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookLocate[\"", "MR"],
  StyleBox["tag", "TI"],
  StyleBox["\"]", "MR"],
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notebook, selecting the cells and scrolling to the position of the first one. \
\n\[FilledSmallSquare] ",
  StyleBox["NotebookLocate[", "MR"],
  Cell[BoxData[
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  StyleBox["\"", "MR"],
  StyleBox["file", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["tag", "TI"],
  StyleBox["\"", "MR"],
  Cell[BoxData[
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  StyleBox["]", "MR"],
  " if necessary opens the notebook stored in ",
  StyleBox["file", "TI"],
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}], "Usage",
  CellTags->{"S0.1329", "NotebookLocate"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
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\[FilledSmallSquare] If the cells are in closed groups, ",
  StyleBox["NotebookLocate", "MR"],
  " will open all these groups. \n\[FilledSmallSquare] ",
  StyleBox["NotebookLocate", "MR"],
  " is used for following hyperlinks within one notebook or between \
notebooks. \n\[FilledSmallSquare] ",
  StyleBox["NotebookLocate", "MR"],
  " searches for tags in the list given as the setting for the ",
  StyleBox["CellTags", "MR"],
  " option of each cell. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.33"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookFind",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookOpen",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["SetSelectedNotebook",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ButtonBox",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookLocate"}],

Cell["NotebookObject", "ObjectName",
  CellTags->{"S0.1329", "NotebookObject"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookObject[", "MR"],
  StyleBox["fe", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["id", "TI"],
  StyleBox["]", "MR"],
  " is an object that represents an open notebook in the front end. "
}], "Usage",
  CellTags->{"S0.1329", "NotebookObject"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["fe", "TI"],
  " is a ",
  StyleBox["FrontEndObject", "MR"],
  " which specifies the front end in which the notebook is open. \n\
\[FilledSmallSquare] ",
  StyleBox["id", "TI"],
  " is an integer that gives a unique serial number for this open notebook. \n\
\[FilledSmallSquare] In ",
  StyleBox["StandardForm", "MR"],
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  StyleBox["OutputForm", "MR"],
  " notebook objects are printed so as to indicate the current title of the \
window that would be used to display the notebook. \n\[FilledSmallSquare] \
Functions such as ",
  StyleBox["NotebookPrint", "MR"],
  " and ",
  StyleBox["NotebookClose", "MR"],
  " take ",
  StyleBox["NotebookObject", "MR"],
  " as their argument. \n\[FilledSmallSquare] Within any open notebook, there \
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applying functions such as ",
  StyleBox["SelectionMove", "MR"],
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  StyleBox["NotebookObject", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookSelection",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookOpen",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Notebooks",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["SelectedNotebook",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookObject"}],

Cell["NotebookOpen", "ObjectName",
  CellTags->{"S0.1329", "NotebookOpen"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookOpen[\"", "MR"],
  StyleBox["name", "TI"],
  StyleBox["\"]", "MR"],
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corresponding notebook object. \n\[FilledSmallSquare] ",
  StyleBox["NotebookOpen[\"", "MR"],
  StyleBox["name", "TI"],
  StyleBox["\",", "MR"],
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  StyleBox["options", "TI"],
  StyleBox["]", "MR"],
  " opens a notebook using the options given. "
}], "Usage",
  CellTags->{"S0.1329", "NotebookOpen"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookOpen", "MR"],
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\[FilledSmallSquare] ",
  StyleBox["NotebookOpen", "MR"],
  " returns ",
  StyleBox["$Failed", "MR"],
  " if it cannot open a notebook with the specified name. \n\
\[FilledSmallSquare] ",
  StyleBox["NotebookOpen", "MR"],
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  StyleBox["NotebookPath", "MR"],
  " global option for the front end. \n\[FilledSmallSquare] With the option \
",
  StyleBox["Visible->False", "MR"],
  " set, ",
  StyleBox["NotebookOpen", "MR"],
  " will return a ",
  StyleBox["NotebookObject", "MR"],
  ", but will not cause a window to appear on your screen. \n\
\[FilledSmallSquare] ",
  StyleBox["NotebookOpen", "MR"],
  " initially sets the current selection to be before the first cell in the \
notebook. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.2",
    ButtonData:>{"2.10.2", "2.16"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.68"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookCreate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookLocate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookSelection",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["OpenRead",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Get",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["SetSelectedNotebook",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookOpen"}],

Cell["NotebookPrint", "ObjectName",
  CellTags->{"S0.1329", "NotebookPrint"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookPrint[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox["]", "MR"],
  " sends a notebook to your printer. \n\[FilledSmallSquare] ",
  StyleBox["NotebookPrint[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["stream", "TI"],
  StyleBox["]", "MR"],
  " sends a PostScript version of the notebook to the specified stream. "
}], "Usage",
  CellTags->{"S0.1329", "NotebookPrint"}],

Cell[TextData[{
  "\[FilledSmallSquare] If ",
  StyleBox["notebook", "TI"],
  " is a ",
  StyleBox["NotebookObject", "MR"],
  ", then ",
  StyleBox["NotebookPrint", "MR"],
  " will print the complete notebook. If it is a ",
  StyleBox["NotebookSelection", "MR"],
  ", then ",
  StyleBox["NotebookPrint", "MR"],
  " will print just the selection. \n\[FilledSmallSquare] ",
  StyleBox["NotebookPrint", "MR"],
  " uses the printing options set for the specified notebook, taking defaults \
from the global options set for the whole front end. \n\[FilledSmallSquare] \
",
  StyleBox["NotebookPrint[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["file", "TI"],
  StyleBox[".ps\"]", "MR"],
  " saves the PostScript form of the notebook in a file. \n\
\[FilledSmallSquare] ",
  StyleBox["NotebookPrint[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["\"!", "MR"],
  StyleBox["command", "TI"],
  StyleBox["\"]", "MR"],
  " gives the PostScript form of the notebook as input to a command. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.68"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookSave",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookWrite",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["PrintingStyleEnvironment",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookPrint"}],

Cell["NotebookPut", "ObjectName",
  CellTags->{"S0.1329", "NotebookPut"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookPut[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " creates a notebook corresponding to ",
  StyleBox["expr", "TI"],
  " and makes it the currently selected notebook in the front end. \n\
\[FilledSmallSquare] ",
  StyleBox["NotebookPut[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["obj", "TI"],
  StyleBox["]", "MR"],
  " replaces the notebook represented by the notebook object ",
  StyleBox["obj", "TI"],
  " with one corresponding to ",
  StyleBox["expr", "TI"],
  ". "
}], "Usage",
  CellTags->{"S0.1329", "NotebookPut"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookPut", "MR"],
  " allows you to take a notebook expression in the kernel and make it an \
open notebook in the front end. \n\[FilledSmallSquare] ",
  StyleBox["expr", "TI"],
  " must be a notebook expression with head ",
  StyleBox["Notebook", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["NotebookPut", "MR"],
  " returns a ",
  StyleBox["NotebookObject", "MR"],
  " corresponding to the notebook it creates. \n\[FilledSmallSquare] ",
  StyleBox["NotebookPut[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["obj", "TI"],
  StyleBox["]", "MR"],
  " overwrites whatever data was contained in the notebook represented by the \
notebook object ",
  StyleBox["obj", "TI"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.2",
    ButtonData:>{"2.10.2", "2.16"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookGet",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookCreate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Put",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookPut"}],

Cell["NotebookRead", "ObjectName",
  CellTags->{"S0.1329", "NotebookRead"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookRead[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox["]", "MR"],
  " gives the expression corresponding to the current selection in the \
specified notebook object. "
}], "Usage",
  CellTags->{"S0.1329", "NotebookRead"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookRead", "MR"],
  " is the basic way to get into the kernel pieces of notebooks that are \
being manipulated by the front end. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.35"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Get",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookWrite",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookDelete",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ButtonSource",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookRead"}],

Cell["Notebooks", "ObjectName",
  CellTags->{"S0.1329", "Notebooks"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Notebooks[", "MR"],
  " ",
  StyleBox["]", "MR"],
  " gives a list of notebooks currently open in the front end. "
}], "Usage",
  CellTags->{"S0.1329", "Notebooks"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Notebooks[", "MR"],
  " ",
  StyleBox["]", "MR"],
  " returns a list of ",
  StyleBox["NotebookObject", "MR"],
  " constructs. \n\[FilledSmallSquare] ",
  StyleBox["Notebooks[", "MR"],
  StyleBox["fe", "TI"],
  StyleBox["]", "MR"],
  " gives a list of notebooks open in a specific front end, specified by a ",
  StyleBox["FrontEndObject", "MR"],
  ". The default is ",
  StyleBox["$FrontEnd", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["SelectedNotebook",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["EvaluationNotebook",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookOpen",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Streams",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "Notebooks"}],

Cell["NotebookSave", "ObjectName",
  CellTags->{"S0.1329", "NotebookSave"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookSave[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox["]", "MR"],
  " saves the current version of a notebook in a file. \n\[FilledSmallSquare] \
",
  StyleBox["NotebookSave[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["file", "TI"],
  StyleBox["\"]", "MR"],
  " saves the notebook in the specified file. \n\[FilledSmallSquare] ",
  StyleBox["NotebookSave[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["stream", "TI"],
  StyleBox["]", "MR"],
  " sends the expression corresponding to the current version of the notebook \
to the specified stream. "
}], "Usage",
  CellTags->{"S0.1329", "NotebookSave"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["notebook", "TI"],
  " must be a ",
  StyleBox["NotebookObject", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["NotebookSave[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox["]", "MR"],
  " saves the notebook in a file whose name is given by the notebook object \
",
  StyleBox["notebook", "TI"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["NotebookSave", "MR"],
  " writes out the ",
  StyleBox["Mathematica", "TI"],
  " expression corresponding to the notebook, together with ",
  StyleBox["Mathematica", "TI"],
  " comments which make it easier for the front end to read the notebook in \
again. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.68"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["NotebookAutoSave",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NotebookPrint",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1329", "NotebookSave"}],

Cell["NotebookSelection", "ObjectName",
  CellTags->{"S0.1329", "NotebookSelection"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookSelection[", "MR"],
  StyleBox["notebook", "TI"],
  StyleBox["]", "MR"],
  " represents the current selection in an open notebook in the front end. "
}], "Usage",
  CellTags->{"S0.1329", "NotebookSelection"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["NotebookSelection", "MR"],
  " takes a ",
  StyleBox["NotebookObject", "MR"],
  " as its argument. \n\[FilledSmallSquare] You can use ",
  StyleBox["Options", "MR"],
  " and ",
  StyleBox["SetOptions", "MR"],
  " to read and write options associated with your current selection. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.3",
    ButtonData:>{"2.10.3", "3.65"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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Multiplying any linear combination of these vectors by the matrix \
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Cell[CellGroupData[{

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  StyleBox["{{\"(\",", "MR"],
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  StyleBox["\")\"},", "MR"],
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  StyleBox["\"\"}", "MR"],
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
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Cell[CellGroupData[{

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Cell["Numerator", "ObjectName",
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["NumericQ", "ObjectName",
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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["O", "ObjectName",
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Cell[TextData[{
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  StyleBox["x", "TI"],
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  StyleBox["n", "TI"],
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  StyleBox["x", "TI"],
  StyleBox[",", "MR"],
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  CellTags->{"S0.1344", "O"}],

Cell[TextData[{
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  StyleBox["The Mathematica Book: ", "TI"],
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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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        Rational[ -1, 4], 
        Rational[ 1, 5], 
        Rational[ -1, 6], 
        Rational[ 1, 7], 
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Cell[BoxData[
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Cell["\<\
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Cell["OddQ", "ObjectName",
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  StyleBox["The Mathematica Book: ", "TI"],
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["Off", "ObjectName",
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\[FilledSmallSquare] ",
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Cell["This prints a warning message. ", "ExampleText",
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[BoxData[
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  CellTags->"Off"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["This switches on tracing of all symbols. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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  CellTags->"Off"]
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[6]:= ",
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Cell[BoxData[
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Cell[BoxData[
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  StyleBox[",", "MR"],
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  StyleBox[",", "MR"],
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
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  StyleBox[",", "MR"],
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\[FilledSmallSquare] ",
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  StyleBox[",", "MR"],
  " ",
  StyleBox["nlist", "TI"],
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  StyleBox["dlist", "TI"],
  StyleBox[",", "MR"],
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  StyleBox[",", "MR"],
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  StyleBox["n", "TI"],
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                SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"x\\\\\\\"\\\",\
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This groups the elements of the original list in pairs; since there \
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              RowBox[{"(", "\[NoBreak]", GridBox[{
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                    {"w", "x"}
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Cell[CellGroupData[{

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Cell[TextData[{
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Cell[TextData[{
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Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell[BoxData[
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Cell["PartitionsP", "ObjectName",
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["Path", "ObjectName",
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Cell[TextData[{
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Cell["Pattern", "ObjectName",
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Cell[TextData[{
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  StyleBox["x:_+_", "MR"],
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  StyleBox["x:(_+_)", "MR"],
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  StyleBox["(x:_)+_", "MR"],
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  StyleBox["s", "TI"],
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  " is equivalent to ",
  StyleBox["s", "TI"],
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Cell["This is the derivative of the generalized polylogarithm. ", \
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",
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Cell["\<\
This gives the greatest common divisor of a pair of polynomials. \
\
\>", "ExampleText",
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Cell[CellGroupData[{

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Cell["Here is the greatest common divisor of another pair. ", "ExampleText",
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        3\ a\^2\ c\^2\ d\^3 - a\^2\ c\^2\ d\^4 + b\^4\ e - 2\ a\ b\^2\ c\ e + 
        a\^2\ c\^2\ e - 2\ b\^4\ d\ e + 6\ a\ b\^2\ c\ d\ e - 
        4\ a\^2\ c\^2\ d\ e + b\^4\ d\^2\ e - 6\ a\ b\^2\ c\ d\^2\ e + 
        6\ a\^2\ c\^2\ d\^2\ e + 2\ a\ b\^2\ c\ d\^3\ e - 
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We can take the gcd of polynomials involving Gaussian rationals, \
and this can even be done modulo a real Gaussian prime. \
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Cell[BoxData[
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Cell[BoxData[
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\[FilledSmallSquare] ",
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  " gives results according to a definite convention; other conventions could \
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We form a Groebner basis from a list of polynomials. When we reduce \
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Cell[TextData[{
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Cell[TextData[{
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Cell["PowerExpand", "ObjectName",
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Cell[TextData[{
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Cell["Further Examples", "ExampleSection",
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Cell["\<\
Fractions are represented internally without either a fraction bar \
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Therefore the first pattern won't work to identify a fraction. \
\
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Cell[BoxData[
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Cell[TextData[{
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",
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Cell["Rationals", "ObjectName",
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Cell[TextData[{
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Cell["Further Examples", "ExampleSection",
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Cell["This number is not rational. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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",
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  CellTags->{"S0.1415", "Raw"}],

Cell["Re", "ObjectName",
  CellTags->{"S0.1416", "Re"}],

Cell[TextData[{
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  StyleBox["]", "MR"],
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  CellTags->{"S0.1416", "Re"}],

Cell[TextData[{
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Cell[CellGroupData[{

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Cell["This extracts the real part of this complex number. ", "ExampleText",
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Cell[TextData[{
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  StyleBox["stream", "TI"],
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  Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["Here is an approximate number. ", "ExampleText",
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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  "A powerful application of ",
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  " is in the partial evaluation of a held expression. Here's a function that \
takes an expression and a list of positions, and evaluates in place the parts \
at the specified positions. "
}], "ExampleText",
  CellTags->"ReplacePart"],

Cell[BoxData[
    \(EvaluateAt[expr_, positions_] := 
      Fold[ReplacePart[#1, Part[#1, Sequence @@ #2], #2] &, expr, 
        positions]\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"ReplacePart"],

Cell[TextData[{
  "In this case, ",
  StyleBox["{1,2}", "MR"],
  " specifies the second element of the held list, and ",
  StyleBox["{1,-1,1}", "MR"],
  " the first part of the last element. "
}], "ExampleText",
  CellTags->"ReplacePart"],

Cell[CellGroupData[{

Cell[BoxData[
    \(EvaluateAt[
      Hold[{2 + 3, 3 + 4, 2*4 + 5}], {{1, 2}, {1, \(-1\), 1}}]\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"ReplacePart"],

Cell[BoxData[
    \(Hold[{2 + 3, 7, 8 + 5}]\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"ReplacePart"]
}, Open  ]],

Cell[BoxData[
    \(Clear[EvaluateAt]\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"ReplacePart"],

Cell["ReplaceRepeated", "ObjectName",
  CellTags->{"S0.1431", "ReplaceRepeated"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["expr", "TI"],
  " ",
  StyleBox["//.", "MR"],
  " ",
  StyleBox["rules", "TI"],
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  StyleBox["expr", "TI"],
  " no longer changes. "
}], "Usage",
  CellTags->{"S0.1431", "ReplaceRepeated"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["expr", "TI"],
  " ",
  StyleBox["//.", "MR"],
  " ",
  StyleBox["rules", "TI"],
  " effectively applies ",
  StyleBox["/.", "MR"],
  " repeatedly, until the results it gets no longer change. \n\
\[FilledSmallSquare] It performs one complete pass over the expression using \
",
  StyleBox["/.", "MR"],
  ", then carries out the next pass. \n\[FilledSmallSquare] You should be \
very careful to avoid infinite loops when you use the ",
  StyleBox["//.", "MR"],
  " operator. The command ",
  StyleBox["x", "MR"],
  " ",
  StyleBox["//.", "MR"],
  " ",
  StyleBox["x", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["x", "MR"],
  " ",
  StyleBox["+", "MR"],
  " ",
  StyleBox["1", "MR"],
  " will, for example, lead to an infinite loop. \n\[FilledSmallSquare] ",
  StyleBox["ReplaceRepeated", "MR"],
  " takes the option ",
  StyleBox["MaxIterations", "MR"],
  ", which specifies the maximum number of times it will try to apply the \
rules you give. The default setting is ",
  StyleBox["MaxIterations", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["65536", "MR"],
  ". With ",
  StyleBox["MaxIterations", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  StyleBox["Infinity", "MR"],
  " there is no limit. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.4.1",
    ButtonData:>{"2.4.1", "1.16"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["ReplaceAll",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Rule",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Set",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FixedPoint",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1431", "ReplaceRepeated"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"ReplaceRepeated"],

Cell[CellGroupData[{

Cell[TextData[{
  "See the Further Examples for ",
  ButtonBox["ReplaceAll",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "ExampleSubsection",
  CellTags->"ReplaceRepeated"],

Cell["ResetDirectory", "ObjectName",
  CellTags->{"S0.1432", "ResetDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["ResetDirectory[", "MR"],
  " ",
  StyleBox["]", "MR"],
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}], "Usage",
  CellTags->{"S0.1432", "ResetDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] Successive calls to ",
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  " uses the directory stack given by ",
  StyleBox["DirectoryStack[", "MR"],
  " ",
  StyleBox["]", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["ResetDirectory", "MR"],
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\[Hyphen]to\[Hyphen]last element current. \n\[FilledSmallSquare] See ",
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  ButtonBox["Section 2.11.4",
    ButtonData:>{"2.11.4", "4.3"},
    ButtonStyle->"MainBookLink"],
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  ButtonBox["Section 2.11.4",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["SetDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Directory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Path",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1432", "ResetDirectory"}],

Cell["Residue", "ObjectName",
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Cell[TextData[{
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  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox["x", "TI"],
  StyleBox[",", "MR"],
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  Cell[BoxData[
      FormBox[
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      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]", "MR"],
  " finds the residue of ",
  StyleBox["expr", "TI"],
  " at the point ",
  Cell[BoxData[
      \(TraditionalForm\`x = x\_0\)], "InlineFormula"],
  ". "
}], "Usage",
  CellTags->{"S0.1433", "Residue"}],

Cell[TextData[{
  "\[FilledSmallSquare] The residue is defined as the coefficient of ",
  StyleBox["(", "MR"],
  StyleBox["x", "TI"],
  " ",
  StyleBox["-", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"x\\\\\\\"\\\", \\\"TI\
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  StyleBox[")^-1", "MR"],
  " in the Laurent expansion of ",
  StyleBox["expr", "TI"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Mathematica", "TI"],
  " can usually find residues at a point only when it can evaluate power \
series at that point. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.6.9",
    ButtonData:>{"3.6.9", "9.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Series",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Limit",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["PrincipalValue",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
  ButtonBox["Algebra`RootIsolation`",
    ButtonData:>{"Algebra`RootIsolation`"},
    ButtonStyle->"AddOnsLink"],
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}], "Notes",
  CellTags->{"S0.1433", "Residue"}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"Residue"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"Residue"],

Cell[BoxData[
    \(1\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"Residue"]
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Cell[CellGroupData[{

Cell[BoxData[
    \(Residue[1\/x\^2, {x, 0}]\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"Residue"],

Cell[BoxData[
    \(0\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"Residue"]
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Cell["Rest", "ObjectName",
  CellTags->{"S0.1434", "Rest"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Rest[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " gives ",
  StyleBox["expr", "TI"],
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}], "Usage",
  CellTags->{"S0.1434", "Rest"}],

Cell[TextData[{
  "\[FilledSmallSquare] Example: ",
  StyleBox["Rest[", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["a, b, c", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]\[LongRightArrow]", "MR"],
  Cell[BoxData[
      \({b, c}\)], "InlineOutput"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Rest[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " is equivalent to ",
  StyleBox["Drop[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["1]", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.8.4",
    ButtonData:>{"1.8.4", "4.7"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Drop",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["First",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Part",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Take",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1434", "Rest"}],

Cell["Resultant", "ObjectName",
  CellTags->{"S0.1435", "Resultant"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Resultant[", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"poly\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"poly\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["var", "TI"],
  StyleBox["]", "MR"],
  " computes the resultant of the polynomials ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"poly\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
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  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"poly\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  " with respect to the variable ",
  StyleBox["var", "TI"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Resultant[", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"poly\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"poly\\\\\\\"\\\", \
\\\"TI\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["var", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["Modulus->", "MR"],
  StyleBox["p", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["p", "TI"],
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}], "Usage",
  CellTags->{"S0.1435", "Resultant"}],

Cell[TextData[{
  "\[FilledSmallSquare] The resultant of two polynomials ",
  Cell[BoxData[
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  " and ",
  Cell[BoxData[
      \(TraditionalForm\`b\)], "InlineFormula"],
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",
  Cell[BoxData[
      \(TraditionalForm\`a\_i - b\_j\)], "InlineFormula"],
  " between roots of the polynomials. The resultant is always a number or a \
polynomial. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.3.4",
    ButtonData:>{"3.3.4", "4.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Subresultants",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["PolynomialGCD",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Eliminate",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1435", "Resultant"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"Resultant"],

Cell[TextData[{
  "Here is the resultant with respect to ",
  StyleBox["y", "MR"],
  " of two polynomials in ",
  StyleBox["x", "MR"],
  " and ",
  StyleBox["y", "MR"],
  ". The original polynomials have a common root in ",
  StyleBox["y", "MR"],
  " only for values of ",
  StyleBox["x", "MR"],
  " at which the resultant vanishes. "
}], "ExampleText",
  CellTags->"Resultant"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Resultant[\((x - y)\)\^2 - 2, y\^2 - 3, y]\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"Resultant"],

Cell[BoxData[
    \(1 - 10\ x\^2 + x\^4\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"Resultant"]
}, Open  ]],

Cell[TextData[{
  "Here ",
  StyleBox["Resultant", "MR"],
  " is used to define ",
  StyleBox["Discriminant", "MR"],
  ". The discriminant of a polynomial is the product of the squares of all \
the differences of the roots taken in pairs. "
}], "ExampleText",
  CellTags->"Resultant"],

Cell[BoxData[
    \(Discriminant[p_, x_] := 
      With[{m = Exponent[p, x]}, 
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\[PartialD]\_x p, x]\)\/Coefficient[p, x, m]]]\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"Resultant"],

Cell["This gives the discriminant for a fifth degree polynomial. ", \
"ExampleText",
  CellTags->"Resultant"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Discriminant[x\^5 - x + a, x]\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"Resultant"],

Cell[BoxData[
    \(\(-256\) + 3125\ a\^4\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"Resultant"]
}, Open  ]],

Cell[BoxData[
    \(Clear[Discriminant]\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"Resultant"],

Cell["Return", "ObjectName",
  CellTags->{"S0.1436", "Return"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Return[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " returns the value ",
  StyleBox["expr", "TI"],
  " from a function. \n\[FilledSmallSquare] ",
  StyleBox["Return[", "MR"],
  " ",
  StyleBox["]", "MR"],
  " returns the value ",
  StyleBox["Null", "MR"],
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}], "Usage",
  CellTags->{"S0.1436", "Return"}],

Cell[TextData[{
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  StyleBox["Return[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " exits control structures within the definition of a function, and gives \
the value ",
  StyleBox["expr", "TI"],
  " for the whole function. \n\[FilledSmallSquare] ",
  StyleBox["Return", "MR"],
  " takes effect as soon as it is evaluated, even if it appears inside other \
functions. \n\[FilledSmallSquare] ",
  StyleBox["Return", "MR"],
  " can be used inside functions like ",
  StyleBox["Scan", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.5.9",
    ButtonData:>{"2.5.9", "9.52"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Break",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Throw",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Abort",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1436", "Return"}],

Cell["Reverse", "ObjectName",
  CellTags->{"S0.1437", "Reverse"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Reverse[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " reverses the order of the elements in ",
  StyleBox["expr", "TI"],
  ". "
}], "Usage",
  CellTags->{"S0.1437", "Reverse"}],

Cell[TextData[{
  "\[FilledSmallSquare] Example: ",
  StyleBox["Reverse[", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["a, b, c", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]\[LongRightArrow]", "MR"],
  Cell[BoxData[
      \({c, b, a}\)], "InlineOutput"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.8.9",
    ButtonData:>{"1.8.9", "9.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Permutations",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["RotateLeft",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["RotateRight",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["StringReverse",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1437", "Reverse"}],

Cell["RGBColor", "ObjectName",
  CellTags->{"S0.1438", "RGBColor"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["RGBColor[", "MR"],
  StyleBox["red", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["green", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["blue", "TI"],
  StyleBox["]", "MR"],
  " is a graphics directive which specifies that graphical objects which \
follow are to be displayed, if possible, in the color given. "
}], "Usage",
  CellTags->{"S0.1438", "RGBColor"}],

Cell[TextData[{
  "\[FilledSmallSquare] Red, green and blue color intensities outside the \
range ",
  StyleBox["0", "MR"],
  " to ",
  StyleBox["1", "MR"],
  " will be clipped. \n\[FilledSmallSquare] On monochrome displays, a gray \
level based on the average of the color intensities is used. \n\
\[FilledSmallSquare] The end of the Graphics Gallery shows examples of colors \
generated with various ",
  StyleBox["RGBColor", "MR"],
  " specifications. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.9.3",
    ButtonData:>{"2.9.3", "3.6"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Hue",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["GrayLevel",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["CMYKColor",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ColorOutput",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
  ButtonBox["Graphics`Colors`",
    ButtonData:>{"Graphics`Colors`"},
    ButtonStyle->"AddOnsLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1438", "RGBColor"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"RGBColor"],

Cell["The second plot distinguishes the curves with colors. ", "ExampleText",
  CellTags->"RGBColor"],

Cell["Evaluate the cells to compare the graphics. ", "Text",
  CellTags->"RGBColor"],

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"RGBColor"],

Cell[BoxData[
    \(Plot[{Sin[x], Sin[2  x], Sin[3  x]}, {x, 0, 2  \[Pi]}, 
      PlotStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 1, 0], 
          RGBColor[0, 0, 1]}]; \)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"RGBColor"],

Cell["RiemannSiegelTheta", "ObjectName",
  CellTags->{"S0.1439", "RiemannSiegelTheta"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["RiemannSiegelTheta[", "MR"],
  StyleBox["t", "TI"],
  StyleBox["]", "MR"],
  " gives the Riemann\[Hyphen]Siegel function ",
  Cell[BoxData[
      \(TraditionalForm\`\[CurlyTheta] \((t)\)\)], "InlineFormula"],
  ". "
}], "Usage",
  CellTags->{"S0.1439", "RiemannSiegelTheta"}],

Cell[TextData[{
  "\[FilledSmallSquare] Mathematical function (see Section ",
  ButtonBox["A.3.10",
    ButtonData:>{"A.3.10", "10.1"},
    ButtonStyle->"MainBookLink"],
  "). \n\[FilledSmallSquare] ",
  Cell[BoxData[
      \(TraditionalForm\`\[CurlyTheta] \((t)\) = 
        "Im" \(\(\[InvisibleSpace]\)\(\([\)\(log  \[CapitalGamma] \((1\/4 + i 
               t\/2)\) - t  log  \[Pi]/2\)\(]\)\)\)\)], "InlineFormula"],
  " for real ",
  Cell[BoxData[
      \(TraditionalForm\`t\)], "InlineFormula"],
  ". \n\[FilledSmallSquare] ",
  Cell[BoxData[
      \(TraditionalForm\`\[CurlyTheta] \((t)\)\)], "InlineFormula"],
  " arises in the study of the Riemann zeta function on the critical line. It \
is closely related to the number of zeros of ",
  Cell[BoxData[
      \(TraditionalForm\`\[Zeta] \((1\/2 + i  u)\)\)], "InlineFormula"],
  " for ",
  Cell[BoxData[
      \(TraditionalForm\`0 < u < t\)], "InlineFormula"],
  ". \n\[FilledSmallSquare] ",
  Cell[BoxData[
      \(TraditionalForm\`\[CurlyTheta] \((t)\)\)], "InlineFormula"],
  " is an analytic function of ",
  Cell[BoxData[
      \(TraditionalForm\`t\)], "InlineFormula"],
  " except for branch cuts on the imaginary axis running from ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[PlusMinus]i/2\)\)], "InlineFormula"],
  " to ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[PlusMinus]i  \[Infinity]\)\)], "InlineFormula"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.2.10",
    ButtonData:>{"3.2.10", "10.17"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["RiemannSiegelZ",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Zeta",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1439", "RiemannSiegelTheta"}],

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  ", where ",
  Cell[BoxData[
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  Cell[BoxData[
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  StyleBox["k", "TI"],
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Cell[BoxData[
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Cell[BoxData[
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It can be cast into the form of a polynomial in one variable. \
\
\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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  CellTags->"Root"],

Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
Here is a more complicated expression involving algebraic numbers. \
\
\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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          142565136\ #1\^7 + 417279801\ #1\^8 + 294900492\ #1\^9 - 
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          92380608\ #1\^13 + 12835545\ #1\^14 - 14722004\ #1\^15 - 
          7159413\ #1\^16 + 970500\ #1\^17 + 2037638\ #1\^18 - 
          380946\ #1\^19 - 301272\ #1\^20 + 61632\ #1\^21 + 49119\ #1\^22 - 
          8292\ #1\^23 - 7811\ #1\^24 + 732\ #1\^25 + 774\ #1\^26 - 
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Cell[TextData[{
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Cell[CellGroupData[{

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}, Open  ]],

Cell["RootReduce", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  ", ",
  ButtonBox["Solve",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
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Cell["Further Examples", "ExampleSection",
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Cell[CellGroupData[{

Cell[TextData[{
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Cell["Roots", "ObjectName",
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Cell[TextData[{
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Multiplicity\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"1\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"multiplicity in final list of solutions \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Quartics\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\
\"MR\\\"]\\) \"\>", \("whether to generate explicit solutions for quartics \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"Using\\\\\\\"\\\", \
\\\"MR\\\"]\\) \"\>", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"True\\\\\\\"\\\", \\\
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Cell[TextData[{
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often given in such cases. \n\[FilledSmallSquare] ",
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than one occur. \n\[FilledSmallSquare] See ",
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    ButtonStyle->"MainBookLink"],
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell["\<\
The roots of some cubic and quartic polynomials can be expressed in \
terms of radicals using Cardano's formulas. The results, while explicit, are \
often very complicated. \
\>", "ExampleText",
  CellTags->"Roots"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Roots"],

Cell[BoxData[
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\@3957)\))\)\^\(1/3\)\/3\^\(2/3\) || 
      x == \(-\(1\/6\)\)\ \((1 + \[ImaginaryI]\ \@3)\)\ \((189\/2 - \(3\ \
\@3957\)\/2)\)\^\(1/3\) - \(\((1 - \[ImaginaryI]\ \@3)\)\ \((1\/2\ \((63 + \
\@3957)\))\)\^\(1/3\)\)\/\(2\ 3\^\(2/3\)\) || 
      x == \(-\(1\/6\)\)\ \((1 - \[ImaginaryI]\ \@3)\)\ \((189\/2 - \(3\ \
\@3957\)\/2)\)\^\(1/3\) - \(\((1 + \[ImaginaryI]\ \@3)\)\ \((1\/2\ \((63 + \
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}, Open  ]],

Cell["You can use algebraic numbers instead. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Roots"],

Cell[BoxData[
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  CellTags->"Roots"]
}, Open  ]],

Cell[TextData[{
  "If any one of the coefficients of the polynomial is inexact, ",
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  " finds approximate roots. "
}], "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"Roots"],

Cell[BoxData[
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      2.2184475681041160595245431807853616499674778`30\ x\^2 + 
      4.1909110038211839073054752861163615306059426`30\ x\^3 - 
      0.9863632647019917876702043670951658867221514`30\ x\^4 + 
      0.661840636697724216248019689303392043021738`30\ x\^5 - 
      8.1137950327455392112357856096334306815301731`30\ x\^6 - 
      3.2313241127735597757989423297009461718355468`30\ x\^7 + 
      2.2302806340244421819968601631553324517647625`30\ x\^8 + 
      3.3283978275836843715533852643296085086907103`30\ x\^9 + 
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Roots[poly == 0, x]\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"Roots"],

Cell[BoxData[
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      x == \(-0.6249668365954184438859712460748195493016816`29.7623\) - 
          0.5126539761908479775236822703846520268687022`29.6763\ \
\[ImaginaryI] || 
      x == \(-0.6249668365954184438859712460748195493016816`29.7623\) + 
          0.5126539761908479775236822703846520268687022`29.6763\ \
\[ImaginaryI] || 
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|| x == \(-0.4224471176784499784822289701406591372148907`29.3693\) + 
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|| x == \(-0.0389933994100204256592595976246138010187367`28.4805\) - 
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\[ImaginaryI] || 
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\[ImaginaryI] || 
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\[ImaginaryI] || 
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\[ImaginaryI] || 
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Cell["\<\
You can find the roots of univariate polynomials for any finite \
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degree of the polynomial. 
This is in contrast to the fundamental theorem of algebra. It states that \
over the complexes there are as many roots (counting multiplicity) as the \
degree. \
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This gives the sum of the seventh powers of the roots of the \
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Cell["\<\
In this polynomial the coefficients of the linear term is symbolic. \
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Cell["SampleDepth", "ObjectName",
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Cell[TextData[{
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Cell["Further Examples", "ExampleSection",
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Cell["Here is a frequency\[Hyphen]modulated tone. ", "ExampleText",
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Cell["Evaluate the cell to see the graphic and hear the sound. ", "Text",
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Cell[BoxData[
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Cell[TextData[{
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Cell["Evaluate the cell to see the graphic and hear the sound. ", "Text",
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Cell[BoxData[
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Cell["SampledSoundFunction", "ObjectName",
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Cell[TextData[{
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  StyleBox["f", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["n", "TI"],
  StyleBox[",", "MR"],
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Cell[TextData[{
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  "."
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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Cell["Evaluate the cell to see the graphic and hear the sound. ", "Text",
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Cell[BoxData[
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Cell["SampledSoundList", "ObjectName",
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Cell[TextData[{
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  Cell[BoxData[
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  Cell[BoxData[
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"a\\\\\\\"\\\", \\\"TI\
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
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  StyleBox["r", "TI"],
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",
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Cell[TextData[{
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell["SetAccuracy", "ObjectName",
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Cell[TextData[{
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Here is a machine\[Hyphen]precision real number approximation of \
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\[FilledSmallSquare] ",
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",
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",
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Cell["\<\
Compare the two plots to see the effect of resetting the option. \
\
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Cell["Evaluate the cells to see the graphics. ", "Text",
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",
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  ButtonBox["Section 1.5.7",
    ButtonData:>{"1.5.7", "7.3"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 3.4.4",
    ButtonData:>{"3.4.4", "4.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] Implementation Notes: see section ",
  ButtonBox["A.9.5",
    ButtonData:>{"A.9.5", "5.7"},
    ButtonStyle->"MainBookLink"],
  ". \n\[FilledSmallSquare] See also: ",
  ButtonBox["Reduce",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Eliminate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["SolveAlways",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Roots",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["NSolve",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FindRoot",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LinearSolve",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["RowReduce",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["GroebnerBasis",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["DSolve",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Root",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related packages: ",
  ButtonBox["Algebra`RootIsolation`",
    ButtonData:>{"Algebra`RootIsolation`"},
    ButtonStyle->"AddOnsLink"],
  ", ",
  ButtonBox["DiscreteMath`RSolve`",
    ButtonData:>{"DiscreteMath`RSolve`"},
    ButtonStyle->"AddOnsLink"],
  ", ",
  ButtonBox["Graphics`ImplicitPlot`",
    ButtonData:>{"Graphics`ImplicitPlot`"},
    ButtonStyle->"AddOnsLink"],
  ", ",
  ButtonBox["Algebra`AlgebraicInequalities`",
    ButtonData:>{"Algebra`AlgebraicInequalities`"},
    ButtonStyle->"AddOnsLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1479", "Solve"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell["Polynomial equations in one variable", "ExampleSubsection",
  CellTags->"Solve"],

Cell["\<\
These are standard formulas for the solutions of normalized \
quadratic and cubic equations. \
\>", "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^2 + 2  b\ x + c == 0, x]\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-b\) - \@\(b\^2 - c\)}, {x \[Rule] \(-b\) + \@\(b\^2 - \
c\)}}\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^3 + q\/3\ x + r\/27 == 0, x]\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-\(\(2\^\(1/3\)\ q\)\/\(3\ \((\(-r\) + \@\(4\ q\^3 + r\^2\
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\(1/3\)\)}, {x \[Rule] \(\((1 + \[ImaginaryI]\ \@3)\)\ q\)\/\(3\ 2\^\(2/3\)\ \
\((\(-r\) + \@\(4\ q\^3 + r\^2\))\)\^\(1/3\)\) - \(\((1 - \[ImaginaryI]\ \@3)\
\)\ \((\(-r\) + \@\(4\ q\^3 + r\^2\))\)\^\(1/3\)\)\/\(6\ 2\^\(1/3\)\)}, {x \
\[Rule] \(\((1 - \[ImaginaryI]\ \@3)\)\ q\)\/\(3\ 2\^\(2/3\)\ \((\(-r\) + \
\@\(4\ q\^3 + r\^2\))\)\^\(1/3\)\) - \(\((1 + \[ImaginaryI]\ \@3)\)\ \
\((\(-r\) + \@\(4\ q\^3 + r\^2\))\)\^\(1/3\)\)\/\(6\ 2\^\(1/3\)\)}}\)], \
"Output",
  CellLabel->"Out[2]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell["\<\
Here are two simple equations of higher degree with solutions in \
terms of powers. They can be rewritten in terms of trigonometric functions \
that sometimes automatically reduce to radicals. \
\>", "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^6 == 1, x]\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"Solve"],

Cell[BoxData[
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\(2/3\)}}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^6 == 1, 
        x] /. \((a_ -> b_)\) :> \((a -> ComplexExpand[b])\)\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-1\)}, {x \[Rule] 
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\[Rule] 1\/2 + \(\[ImaginaryI]\ \@3\)\/2}, {x \[Rule] 
          1\/2 - \(\[ImaginaryI]\ \@3\)\/2}, {x \[Rule] \(-\(1\/2\)\) + \(\
\[ImaginaryI]\ \@3\)\/2}}\)], "Output",
  CellLabel->"Out[4]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^7 == 1, x]\)], "Input",
  CellLabel->"In[5]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] 
          1}, {x \[Rule] \(-\((\(-1\))\)\^\(1/7\)\)}, {x \[Rule] \((\(-1\))\)\
\^\(2/7\)}, {x \[Rule] \(-\((\(-1\))\)\^\(3/7\)\)}, {x \[Rule] \((\(-1\))\)\^\
\(4/7\)}, {x \[Rule] \(-\((\(-1\))\)\^\(5/7\)\)}, {x \[Rule] \
\((\(-1\))\)\^\(6/7\)}}\)], "Output",
  CellLabel->"Out[5]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^7 == 1, 
        x] /. \((a_ -> b_)\) :> \((a -> ComplexExpand[b])\)\)], "Input",
  CellLabel->"In[6]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] 
          1}, {x \[Rule] \(-Cos[\[Pi]\/7]\) - \[ImaginaryI]\ Sin[\[Pi]\/7]}, \
{x \[Rule] 
          Cos[\(2\ \[Pi]\)\/7] + \[ImaginaryI]\ Sin[\(2\ \[Pi]\)\/7]}, {x \
\[Rule] \(-Cos[\(3\ \[Pi]\)\/7]\) - \[ImaginaryI]\ Sin[\(3\ \[Pi]\)\/7]}, {x \
\[Rule] Cos[\(4\ \[Pi]\)\/7] + \[ImaginaryI]\ Sin[\(4\ \[Pi]\)\/7]}, {x \
\[Rule] \(-Cos[\(5\ \[Pi]\)\/7]\) - \[ImaginaryI]\ Sin[\(5\ \[Pi]\)\/7]}, {x \
\[Rule] Cos[\(6\ \[Pi]\)\/7] + \[ImaginaryI]\ Sin[\(6\ \[Pi]\)\/7]}}\)], \
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  CellLabel->"Out[6]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "For some equations, ",
  StyleBox["Mathematica", "TI"],
  " produces the result in terms of ",
  StyleBox["Root", "MR"],
  " objects (or algebraic numbers). The first argument of ",
  StyleBox["Root", "MR"],
  " is an irreducible polynomial expressed as a pure function and the second \
argument identifies the choice of root. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^5 - 1331  x + 11 == 0, x]\)], "Input",
  CellLabel->"In[7]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] Root[11 - 1331\ #1 + #1\^5 &, 1]}, {x \[Rule] 
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          Root[11 - 1331\ #1 + #1\^5 &, 3]}, {x \[Rule] 
          Root[11 - 1331\ #1 + #1\^5 &, 4]}, {x \[Rule] 
          Root[11 - 1331\ #1 + #1\^5 &, 5]}}\)], "Output",
  CellLabel->"Out[7]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "We can get a numerical result by applying ",
  StyleBox["N", "MR"],
  ". "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(N[%]\)], "Input",
  CellLabel->"In[8]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-6.04216970578258`\)}, {x \[Rule] 
          0.008264462809946322`}, {x \[Rule] 
          6.038037469542513`}, {x \[Rule] \(-0.002066113284939944`\) - 
            6.040107121403968`\ \[ImaginaryI]}, {x \[Rule] \
\(-0.002066113284939944`\) + 6.040107121403968`\ \[ImaginaryI]}}\)], "Output",\

  CellLabel->"Out[8]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "This pulls all of the rules for ",
  StyleBox["x", "MR"],
  " out of the result. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(x /. %\)], "Input",
  CellLabel->"In[9]:= ",
  CellTags->"Solve"],

Cell[BoxData[
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  CellLabel->"Out[9]= ",
  CellTags->"Solve"]
}, Open  ]]
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Cell[CellGroupData[{

Cell["Polynomial equations in more than one variable", "ExampleSubsection",
  CellTags->"Solve"],

Cell[TextData[{
  "Here we solve for ",
  StyleBox["x", "MR"],
  " and ",
  StyleBox["y", "MR"],
  ". "
}], "ExampleText",
  CellTags->"Solve"],

Cell[BoxData[
    \(eqns = {x == 1 + 2\ a\ y, y == 9 + 2\ x}; \)], "Input",
  CellLabel->"In[10]:= ",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[eqns, {x, y}]\)], "Input",
  CellLabel->"In[11]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-\(\(1 + 18\ a\)\/\(\(-1\) + 4\ a\)\)\), 
        y \[Rule] \(-\(11\/\(\(-1\) + 4\ a\)\)\)}}\)], "Output",
  CellLabel->"Out[11]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "In this case we eliminate ",
  StyleBox["y", "MR"],
  " first and then solve for ",
  StyleBox["x", "MR"],
  ". "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[eqns, x, y]\)], "Input",
  CellLabel->"In[12]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-\(\(1 + 18\ a\)\/\(\(-1\) + 4\ a\)\)\)}}\)], "Output",
  CellLabel->"Out[12]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[BoxData[
    \(Clear[eqns]\)], "Input",
  CellLabel->"In[13]:= ",
  CellTags->"Solve"],

Cell[TextData[{
  "Here we solve two simultaneous algebraic equations. The spurious potential \
solution ",
  Cell[BoxData[
      \(y -> 1\)], "InlineFormula"],
  " is rejected. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[{\(1 + x\)\/\(\(-1\) + y\) == 0, 2 - 3  y + y\^2 == 0}, {x, 
        y}]\)], "Input",
  CellLabel->"In[14]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-1\), y \[Rule] 2}}\)], "Output",
  CellLabel->"Out[14]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "Here are three simultaneous algebraic equations; ",
  StyleBox["y", "MR"],
  " and ",
  StyleBox["z", "MR"],
  " must be paired up correctly with ",
  StyleBox["x", "MR"],
  ". "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[{1 + x\^2 == 0, y\^2 == x\^2, z == x}, {x, y, z}]\)], "Input",
  CellLabel->"In[15]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{y \[Rule] \(-\[ImaginaryI]\), z \[Rule] \(-\[ImaginaryI]\), 
        x \[Rule] \(-\[ImaginaryI]\)}, {y \[Rule] \(-\[ImaginaryI]\), 
        z \[Rule] \[ImaginaryI], 
        x \[Rule] \[ImaginaryI]}, {y \[Rule] \[ImaginaryI], 
        z \[Rule] \(-\[ImaginaryI]\), 
        x \[Rule] \(-\[ImaginaryI]\)}, {y \[Rule] \[ImaginaryI], 
        z \[Rule] \[ImaginaryI], x \[Rule] \[ImaginaryI]}}\)], "Output",
  CellLabel->"Out[15]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "Here ",
  StyleBox["Solve", "MR"],
  " returns an empty list, indicating no solution. Every potential solution \
forces an equation in the parameter ",
  StyleBox["z", "MR"],
  " alone, so there are no generic solutions. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[{\((x - y)\)\ z\^2 == 1, x\^2 == 3, y\^2 == 3}, {x, 
        y}]\)], "Input",
  CellLabel->"In[16]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({}\)], "Output",
  CellLabel->"Out[16]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "We can get solutions by solving for ",
  StyleBox["z", "MR"],
  " as well as ",
  StyleBox["x", "MR"],
  " and ",
  StyleBox["y", "MR"],
  ". "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[{\((x - y)\)\ z\^2 == 1, x\^2 == 3, y\^2 == 3}, {x, y, 
        z}]\)], "Input",
  CellLabel->"In[17]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-\@3\), y \[Rule] \@3, 
        z \[Rule] \(-\(\[ImaginaryI]\/\(\@2\ 3\^\(1/4\)\)\)\)}, {x \[Rule] \
\(-\@3\), y \[Rule] \@3, 
        z \[Rule] \[ImaginaryI]\/\(\@2\ 3\^\(1/4\)\)}, {x \[Rule] \@3, 
        y \[Rule] \(-\@3\), 
        z \[Rule] \(-\(1\/\(\@2\ 3\^\(1/4\)\)\)\)}, {x \[Rule] \@3, 
        y \[Rule] \(-\@3\), z \[Rule] 1\/\(\@2\ 3\^\(1/4\)\)}}\)], "Output",
  CellLabel->"Out[17]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "We eliminate ",
  StyleBox["e", "MR"],
  ", ",
  StyleBox["s", "MR"],
  ", and ",
  StyleBox["t", "MR"],
  " to get ",
  StyleBox["d", "MR"],
  " in terms of the remaining variables. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[{1\/s + 1\/t == 1\/F, 1\/\(d + s\) + 1\/\(t - e\) == 1\/F, 
        c == \(e\ F\)\/\(f\ \((t - e)\)\), M == t\/s}, 
      d, {e, s, t}]\)], "Input",
  CellLabel->"In[18]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{d \[Rule] \(-\(\(c\ f\ F\ \((1 + M)\)\)\/\(M\ \((c\ f - 
                      F\ M)\)\)\)\)}}\)], "Output",
  CellLabel->"Out[18]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell["\<\
Verification is not done by default for polynomial systems. The \
purported solutions to this system are nowhere near correct. \
\>", \
"ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(poly1 = \(-49.83328064552891\) + 703.3295204301365  x\^2 + 
        1022.781090460273  x\ y + 895.055397230137  y\^2; \n\n
    poly2 = \(-54.89903052623909\) + 791.4603546519813  x\^2 + 
        1150.940862103963  x\ y + 959.135283051981  y\^2; \n\n{poly1, poly2} /. 
      Solve[{poly1, poly2} == 0, {x, y}]\)], "Input",
  CellLabel->"In[19]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{\(-36.99599256180139`\), \(-41.63178217765983`\)}, \
{\(-36.99599256180139`\), \(-41.63178217765983`\)}, \
{\(-37.005549225841534`\), \(-41.642536341235775`\)}, {\(-37.005549225841534`\
\), \(-41.642536341235775`\)}}\)], "Output",
  CellLabel->"Out[19]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "When there is numeric instability, setting ",
  StyleBox["VerifySolutions", "MR"],
  " to ",
  StyleBox["True", "MR"],
  " will take longer but will make the result more reliable. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \({poly1, poly2} /. 
      Solve[{poly1, poly2} == 0, {x, y}, 
        VerifySolutions \[Rule] True]\)], "Input",
  CellLabel->"In[20]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{0.`, 0.`}, {3.552713678800501`*^-15, 
        3.552713678800501`*^-15}, {3.552713678800501`*^-15, 
        3.552713678800501`*^-15}, {0.`, 0.`}}\)], "Output",
  CellLabel->"Out[20]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[BoxData[
    \(Clear[poly1, poly2]\)], "Input",
  CellLabel->"In[21]:= ",
  CellTags->"Solve"]
}, Closed]],

Cell[CellGroupData[{

Cell["Radical equations", "ExampleSubsection",
  CellTags->"Solve"],

Cell[TextData[{
  "In radical equations, ",
  StyleBox["Solve", "MR"],
  " discards parasite solutions. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[1\/x + 1\/\@x == 1, x]\)], "Input",
  CellLabel->"In[22]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] 1\/2\ \((3 + \@5)\)}}\)], "Output",
  CellLabel->"Out[22]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x\^\(1/3\) + \@x == 1, x]\)], "Input",
  CellLabel->"In[23]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] \(-\(2\/3\)\) - 
            11\/3\ \((2\/\(101 + 15\ \@69\))\)\^\(1/3\) + 
            1\/3\ \((1\/2\ \((101 + 15\ \@69)\))\)\^\(1/3\)}}\)], "Output",
  CellLabel->"Out[23]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[x + x\^\(1/2\) + x\^\(1/3\) == 5, x]\)], "Input",
  CellLabel->"In[24]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] 
          Root[15625 - 20875\ #1 + 11071\ #1\^2 - 3005\ #1\^3 + 440\ #1\^4 - 
                33\ #1\^5 + #1\^6 &, 1]}}\)], "Output",
  CellLabel->"Out[24]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "To see all candidate solutions, including parasites, set ",
  StyleBox["VerifySolutions", "MR"],
  " to ",
  StyleBox["False", "MR"],
  ". "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[1\/x + 1\/\@x == 1, VerifySolutions \[Rule] False]\)], "Input",
  CellLabel->"In[25]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({{x \[Rule] 1\/2\ \((3 - \@5)\)}, {x \[Rule] 
          1\/2\ \((3 + \@5)\)}}\)], "Output",
  CellLabel->"Out[25]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[TextData[{
  "If you can be sure that zero denominators will not affect the solution \
set, clearing them often makes ",
  StyleBox["Solve", "MR"],
  " faster. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(expr = \(-\(\(k - x\)\/\@\(b\^2 + \((k - x)\)\^2\)\)\) + 
        x\/\@\(a\^2 + x\^2\); \n\nTiming[Solve[expr == 0, x]]\)], "Input",
  CellLabel->"In[26]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({0.28000000000000114`\ Second, {{x \[Rule] \(a\ k\)\/\(a - b\)}, {x \
\[Rule] \(a\ k\)\/\(a + b\)}}}\)], "Output",
  CellLabel->"Out[26]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Timing[Solve[Numerator[Together[expr]] == 0, x]]\)], "Input",
  CellLabel->"In[27]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \({0.05999999999997385`\ Second, {{x \[Rule] \(a\ k\)\/\(a - b\)}, {x \
\[Rule] \(a\ k\)\/\(a + b\)}}}\)], "Output",
  CellLabel->"Out[27]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell[BoxData[
    \(expr =  . \)], "Input",
  CellLabel->"In[28]:= ",
  CellTags->"Solve"]
}, Closed]],

Cell[CellGroupData[{

Cell["Suppressing messages", "ExampleSubsection",
  CellTags->"Solve"],

Cell[TextData[{
  "The equations that follow in the rest of this notebook come from various \
papers and from questions submitted by users. \n",
  StyleBox["Solve", "MR"],
  " generates warning messages for many non\[Hyphen]polynomial equations. "
}], "ExampleText",
  CellTags->"Solve"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[\[ExponentialE]\^z == w\ z\^\(-n\), z]\)], "Input",
  CellLabel->"In[29]:= ",
  CellTags->"Solve"],

Cell[BoxData[
    \(InverseFunction::"ifun" \(\(:\)\(\ \)\) 
      "Inverse functions are being used. Values may be lost for multivalued \
inverses."\)], "Message",
  CellTags->"Solve"],

Cell[BoxData[
    \(InverseFunction::"ifun" \(\(:\)\(\ \)\) 
      "Inverse functions are being used. Values may be lost for multivalued \
inverses."\)], "Message",
  CellTags->"Solve"],

Cell[BoxData[
    \(Solve::"ifun" \(\(:\)\(\ \)\) 
      "Inverse functions are being used by \!\(Solve\), so some solutions may \
not be found."\)], "Message",
  CellTags->"Solve"],

Cell[BoxData[
    \({{z \[Rule] n\ ProductLog[w\^\(1\/n\)\/n]}}\)], "Output",
  CellLabel->"Out[29]= ",
  CellTags->"Solve"]
}, Open  ]],

Cell["\<\
To improve readability, we will now suppress warning messages about \
inverse functions. These messages will be turned back on at the end of this \
set of examples. \
\>", "ExampleText",
  CellTags->"Solve"],

Cell[BoxData[
    \(Off[Solve::"\<ifun\>"]; Off[InverseFunction::"\<ifun\>"]; \)], "Input",
  CellLabel->"In[30]:= ",
  CellTags->"Solve"]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Equations involving trigonometric or hyperbolic functions, or their \
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\>", "ExampleSubsection",
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["Equations involving exponentials and logarithms", "ExampleSubsection",
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
We will now turn the inverse function warning messages back on that \
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Cell["Further Examples", "ExampleSection",
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Cell["Here is a simple example. ", "ExampleText",
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Cell["This fails because it involves transcendentals. ", "ExampleText",
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Cell[CellGroupData[{

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Cell["We check the result. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Both of these commands sort the list from largest to smallest. \
\
\>", "ExampleText",
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["Here are two numbers. ", "ExampleText",
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
Ordering numeric expressions according to magnitude by default \
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\>", \
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
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Cell[BoxData[
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Cell["SphericalHarmonicY", "ObjectName",
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Cell[BoxData[
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Cell["These are the partial derivatives. ", "ExampleText",
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell["SphericalRegion", "ObjectName",
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Cell[TextData[{
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Cell["Splice", "ObjectName",
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Cell[TextData[{
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Cell[BoxData[
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Cell[TextData[{
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Cell[BoxData[
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\[FilledSmallSquare] ",
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\[FilledSmallSquare] See ",
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Cell["Sqrt", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["Sqrt[", "MR"],
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\[FilledSmallSquare] In ",
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Cell[BoxData[
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Cell["\<\
The output of this function depends on the type of the input. \
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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["SymbolName", "ObjectName",
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Cell[TextData[{
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}], "Notes",
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Cell["SyntaxLength", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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\[FilledSmallSquare] If ",
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Cell["SyntaxQ", "ObjectName",
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Cell[TextData[{
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  CellTags->{"S0.1521", "TensorRank"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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  Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[1]= ",
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Cell["This is the tensor rank of a 2 x 4 x 1 tensor. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[2]= ",
  CellTags->"TensorRank"]
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Cell["TeXForm", "ObjectName",
  CellTags->{"S0.1521", "TeXForm"}],

Cell[TextData[{
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  StyleBox["TeXForm[", "MR"],
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
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  " and ",
  StyleBox["ALPHA", "MR"],
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\[FilledSmallSquare] All standard ",
  StyleBox["Mathematica", "TI"],
  " box structures are translated by ",
  StyleBox["TeXForm", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["Mathematica", "TI"],
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  ButtonBox["StandardForm",
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  ButtonBox["TraditionalForm",
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}], "Notes",
  CellTags->{"S0.1521", "TeXForm"}]
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"TeXForm"],

Cell["Here are two expressions converted to TeX form. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"TeXForm"],

Cell["{\\sqrt{x^2 + y^2}}", "Output",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"TeXForm"],

Cell["\\sin (\\frac{\\pi }{17})", "Output",
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  CellTags->"TeXForm"]
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Cell["TeXSave", "ObjectName",
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Cell[TextData[{
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  StyleBox["TeXSave[\"", "MR"],
  StyleBox["file", "TI"],
  StyleBox[".tex\"]", "MR"],
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  StyleBox[".tex\",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["source", "TI"],
  StyleBox[".nb\"]", "MR"],
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  StyleBox["source", "TI"],
  StyleBox[".nb", "MR"],
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  StyleBox["TeXSave[\"", "MR"],
  StyleBox["file", "TI"],
  StyleBox[".tex\",", "MR"],
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  StyleBox["notebook", "TI"],
  StyleBox["]", "MR"],
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  CellTags->{"S0.1521", "TeXSave"}],

Cell[TextData[{
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\[FilledSmallSquare] ",
  StyleBox["TeXSave", "MR"],
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  CellTags->{"S0.1521", "TeXSave"}]
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Cell[CellGroupData[{

Cell["Additional Information", "ExampleSection",
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Cell["User Interface", "Subsubsection",
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Cell[TextData[{
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  StyleBox["Mathematica", "TI"],
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  StyleBox["TeXSave", "MR"],
  ", which may be accessed either directly or from the ",
  StyleBox["Save As Special", "SR"],
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}], "Text",
  CellTags->"TeXSave"],

Cell["Required Materials for Processing Output", "Subsubsection",
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Cell[TextData[{
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  Cell[BoxData[
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  StyleBox["TeXSave", "MR"],
  " output employs specialized macros, which are defined in the LaTeX package \
",
  StyleBox["notebook2e.sty", "MR"],
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  StyleBox["Mathematica", "TI"],
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  StyleBox["notebook2e.sty", "MR"],
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instead of the TeX\[Hyphen]native Computer Modern fonts. Your LaTeX \
installation will need to be configured to support the creation of DVI files \
that use these fonts. It is unlikely that you will need to make any \
adjustments since the default setup of most current TeX distributions \
includes this feature. If you performed a minimal installation of the \
distribution, you may need to install additional modules for that \
distribution. Refer to your distribution's Local Guide for more information. \
\nThe math mode macros in ",
  StyleBox["notebook2e.sty", "MR"],
  " typeset special characters using the same fonts used by ",
  StyleBox["Mathematica", "TI"],
  ". This is done with the help of Jens\[Hyphen]Peer Kuska's ",
  StyleBox["Mathematica", "TI"],
  " Virtual Font Package. Since virtual fonts are used instead of PK fonts, \
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The characters will be visible after the DVI file has been converted to \
PostScript by a driver like DVIPS. \nThe ",
  StyleBox["notebook2e.sty", "MR"],
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in comments at the beginning of the ",
  StyleBox["Mathematica", "TI"],
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as straightforward as possible. \nOnce the required files are copied to your \
TeX installation, you may need to refresh your TeX installation's database of \
filenames so that TeX can find the new package files. Consult your TeX \
distribution's documentation for specific instructions on doing this. \
Additionally, you will need to configure your DVI to PostScript driver to \
find the Type 1 versions of the Math fonts on your system. \n",
  StyleBox["Mathematica", "TI"],
  " licensees are permitted to supply copies of the ",
  StyleBox[".sty", "MR"],
  " files and the Type 1 versions of the Math fonts to publishers and service \
bureaus for processing files generated by ",
  StyleBox["TeXSave", "MR"],
  ". You can obtain clarification on copyright issues associated with these \
files by sending e\[Hyphen]mail to permissions@wolfram.com. "
}], "Text",
  CellTags->"TeXSave"],

Cell["Principles of Notebook Conversion", "Subsubsection",
  CellTags->"TeXSave"],

Cell[TextData[{
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  StyleBox["TeXSave", "MR"],
  " separates content from formatting details by treating ",
  StyleBox["Mathematica", "TI"],
  " cell styles as abstractions. Little explicit formatting information is \
retained in the conversion of a plain text cell. "
}], "Text",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell[TextData[{
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  StyleBox["Subsection", "MR"],
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}], "ExampleText",
  CellTags->"TeXSave"],

Cell["\<\
Cell[TextData[\"Survey of existing literature\"], \"Subsection\"]\
\
\>", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell["\<\
This is the output written to the TeX input file after conversion. \
The output does not explicitly specify the font choice or how the text should \
be positioned within the document. \
\>", "ExampleText",
  CellTags->"TeXSave"],

Cell["\\Subsection{Survey of existing literature}", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[TextData[{
  "For cells that contain two\[Hyphen]dimensional typeset math expressions, \
",
  StyleBox["TeXSave", "MR"],
  " generates custom macro names based on the cell's environment, format, and \
cell style. "
}], "Text",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell[TextData[{
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  StyleBox["Text", "MR"],
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  StyleBox["TraditionalForm", "MR"],
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}], "ExampleText",
  CellTags->"TeXSave"],

Cell["\<\
Cell[TextData[{
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  Cell[BoxData[
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              RowBox[{\"2\", \" \"}]], \"+\", \" \",
            SuperscriptBox[\"b\", \"2\"]}]}], TraditionalForm]]],
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Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell[TextData[{
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provided with ",
  StyleBox["Mathematica", "TI"],
  ". The macro \[Backslash]",
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Cell["\<\
The Pythagorean theorem states that \\inlineTFinmath{{c^2}=\\multsp
{a^{2\\multsp }}+\\multsp {b^2}}.\
\>", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[TextData[{
  "The control over the appearance of the text resides within the macro \
definitions in the LaTeX2",
  Cell[BoxData[
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file, ",
  StyleBox["notebook2e.sty", "MR"],
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  Cell[BoxData[
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  StyleBox["notebook.sty", "MR"],
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  StyleBox["Mathematica", "TI"],
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specifying an additional custom package file that contains private macro \
definitions and modifications. \nA large number of cell styles are supported \
by ",
  StyleBox["TeXSave", "MR"],
  " and the ",
  StyleBox[".sty", "MR"],
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  StyleBox["Mathematica", "TI"],
  ". If you use a style sheet that is not part of the ",
  StyleBox["Mathematica", "TI"],
  " distribution, you may need to add additional definitions for macros based \
on the new style names. "
}], "Text",
  CellTags->"TeXSave"],

Cell["Options for Adjusting Output", "Subsubsection",
  CellTags->"TeXSave"],

Cell[TextData[{
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  StyleBox["TeXSave", "MR"],
  " can be adjusted through ",
  StyleBox["ConversionOptions", "MR"],
  ", which takes a list of suboptions as its value. \nThe suboption ",
  StyleBox["\"TeXVersion\"", "MR"],
  " specifies which version of LaTeX the should be used in generating output. \
The default value is ",
  StyleBox["\"LaTeX2e\"", "MR"],
  ". Output that is compatible with LaTeX 2.09 can be obtained by setting the \
value to ",
  StyleBox["\"LaTeX2.09\"", "MR"],
  ". Due to inherent limitations in the font selection mechanism of LaTeX \
2.09, the macros in ",
  StyleBox["notebook.sty", "MR"],
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and ",
  StyleBox["TraditionalForm", "MR"],
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output in ",
  StyleBox["InputForm", "MR"],
  ", ",
  StyleBox["OutputForm", "MR"],
  ", and ",
  StyleBox["StandardForm", "MR"],
  ". \nThe ",
  StyleBox["\"DocumentClass\"", "MR"],
  " suboption specifies what type of document style LaTeX should use when \
processing the TeX input file. Typical values for this option are ",
  StyleBox["\"article\"", "MR"],
  " and ",
  StyleBox["\"report\"", "MR"],
  ". \nWhere possible, ",
  StyleBox["TeXSave", "MR"],
  " will convert graphics within a notebook to Encapsulated PostScript \
format. The option ",
  StyleBox["\"EmbeddedGraphics\"", "MR"],
  " specifies where to direct the EPS output. When the option is set to ",
  StyleBox["False", "MR"],
  ", each graphic will be written to an individual EPS file with the name ",
  StyleBox["filename", "MO"],
  Cell[BoxData[
      \(TraditionalForm\`\ \)], "InlineFormula"],
  StyleBox["_gr", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`\ \)], "InlineFormula"],
  StyleBox["number", "MO"],
  StyleBox[".eps", "MR"],
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  StyleBox["True", "MR"],
  ", the EPS will be included within the TeX input file. The EPS does not \
contain a bitmap preview, so the graphic will not be rendered if your TeX \
preview software does not have a built\[Hyphen]in PostScript interpreter. The \
graphics will be rendered properly when printed with a PostScript printer or \
displayed on a full\[Hyphen]featured PostScript onscreen previewer. \nThe \
correspondence between ",
  StyleBox["Mathematica", "TI"],
  " cell styles and TeX macros is determined by the settings in the option ",
  StyleBox["\"MarkupRules\"", "MR"],
  ". If this option is not set explicitly by the user, ",
  StyleBox["TeXSave", "MR"],
  " will use its own built\[Hyphen]in conversion rules. The value of this \
option is a list of items. Possible items include style sheet notebooks, \
notebook objects, file names specified as strings or as ",
  StyleBox["FrontEnd`FileName", "MR"],
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  StyleBox["ConversionRules", "MR"],
  " options. If two conversion rules specify different settings for the same \
cell style or format type, then the last conversion rule listed will be used. \
\nIf a style sheet notebook is specified, the only options extracted from the \
style definitions are ",
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since it is assumed that details about the appearance of the style will be \
governed by the macro definition that is associated with the style. \nA text \
conversion rule conveys information about what TeX macro should be used to \
delimit the converted cell's contents. Its general structure takes the form \
",
  StyleBox["ConversionRules", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["\"TeX\"", "MR"],
  " ",
  StyleBox["-> ", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["\"", "MR"],
  StyleBox["opening", "TI"],
  " ",
  StyleBox["delimiter", "TI"],
  StyleBox["\",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["closing", "TI"],
  " ",
  StyleBox["delimiter", "TI"],
  StyleBox["\"", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  Cell[BoxData[
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}], "Text",
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Cell[" ", "ProgramTop",
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Cell[TextData[{
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\[Hyphen]defined cell style called ",
  StyleBox["Digression", "MR"],
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}], "ExampleText",
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Cell["\<\
Cell[StyleData[\"Digression\"],
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     ConversionRules -> {
       \"TeX\" -> {\"\\Digression{\", \"}\\n\"}
     }
]\
\>", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell[TextData[{
  "Here is a cell expression that uses the ",
  StyleBox["Digression", "MR"],
  " style."
}], "ExampleText",
  CellTags->"TeXSave"],

Cell["\<\
Cell[TextData[\"Functions defined on \[DoubleStruckCapitalZ]\"],
\"Digression\"]\
\>", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell["\<\
Here is the appearance of the converted cell in the LaTeX input \
file.\
\>", "ExampleText",
  CellTags->"TeXSave"],

Cell["\\Digression{Functions defined on \\DoubleStruckCapitalZ{}}", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[TextData[{
  "For cell styles associated with typeset mathematics, a distinction must be \
made as to whether the typeset expression appears in an inline or a \
freestanding environment. The conversion rules used here have two parts, \
taking the form ",
  StyleBox["ConversionRules", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["\"TeX\"", "MR"],
  " ",
  StyleBox["->", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["inline", "TI"],
  " ",
  StyleBox["formatting", "TI"],
  " ",
  StyleBox["list", "TI"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[", ", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["display", "TI"],
  " ",
  StyleBox["formatting", "TI"],
  " ",
  StyleBox["list", "TI"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  ". Each formatting list contains the following information ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["opening", "TI"],
  " ",
  StyleBox["delimiter", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["formatting", "TI"],
  " ",
  StyleBox["function", "TI"],
  StyleBox[", ", "MR"],
  StyleBox["closing", "TI"],
  " ",
  StyleBox["delimiter", "TI"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  ". For the cases where ",
  StyleBox["TeXSave", "MR"],
  " generates a custom TeX macro, the delimiters specified in the conversion \
rule are overridden, leaving only the formatting function to have an impact \
on the output. A typical formatting function used in conversion rules is ",
  StyleBox["TeXForm", "MR"],
  ". "
}], "Text",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell[TextData[{
  "Here is a cell style definition that contains a conversion rule for a user\
\[Hyphen]defined style called ",
  StyleBox["EmphasizedFormula", "MR"],
  ". "
}], "ExampleText",
  CellTags->"TeXSave"],

Cell["\<\
Cell[StyleData[\"EmphasizedFormula\"],
     other cell options, ...
     ConversionRules -> {
       \"TeX\" -> {
         {\"\\\\inlineEmphasizedFormula{\",
          ToString[#, TeXForm]\\&,
          \"\\}\"
         },
         {\"\\\\displayEmphasizedFormula{\\\\n\",
          ToString[#, TeXForm]\\&,
          \"\\n\\}\\n\"
         }
       }
     }
]\
\>", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[" ", "ProgramTop",
  CellTags->"TeXSave"],

Cell[TextData[{
  "Here is a combination of the prior examples using the ",
  StyleBox["\"MarkupRules\"", "MR"],
  " option hat could be supplied as an argument to ",
  StyleBox["TeXSave", "MR"],
  ". "
}], "ExampleText",
  CellTags->"TeXSave"],

Cell["\<\
\"MarkupRules\" -> {\"Digression\" -> {\"\\\\Digression{\", \
\"}\\n\"},
                  \"EmphasizedFormula\" -> {
                    \"\\\\inlineEmphasizedFormula{\",
                     ToString[#, FormatType->TeXForm]&,
                     \"\\}\"
                    },
                    {\"\\\\displayEmphasizedFormula{\\n\",
                     ToString[#, FormatType -> TeXForm]&,
                     \"\\n\\}\\n\"
                    }
                 }\
\>", "Program",
  CellTags->"TeXSave"],

Cell[" ", "ProgramBottom",
  CellTags->"TeXSave"],

Cell[TextData[{
  "By default, ",
  StyleBox["TeXSave", "MR"],
  " uses the built\[Hyphen]in function ",
  StyleBox["TeXForm", "MR"],
  " to convert typeset math expressions into TeX syntax. If you have set ",
  StyleBox["\"TeXVersion\"", "MR"],
  " to ",
  StyleBox["\"LaTeX2.09\"", "MR"],
  " and desire an appearance that is closer to what you see in a ",
  StyleBox["Mathematica", "TI"],
  " notebook, you can change the option ",
  StyleBox["\"DisplayMathToTeX\"", "MR"],
  " from its default value of ",
  StyleBox["True", "MR"],
  " to MVFalse. ",
  StyleBox["TeXSave", "MR"],
  " will convert each display math formula into an Encapsulated PostScript \
graphic using a file naming system similar to that of files generated by \
converting ",
  StyleBox["Graphics", "MR"],
  " style cells. "
}], "Text",
  CellTags->"TeXSave"],

Cell["Source Code Availability", "Subsubsection",
  CellTags->"TeXSave"],

Cell[TextData[{
  "Despite the large number of options for adjusting ",
  StyleBox["TeXSave", "MR"],
  " output, there may be circumstances where you may need to make changes at \
a deeper level. Since the bulk of ",
  StyleBox["TeXSave", "MR"],
  " is implemented as top\[Hyphen]level ",
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Cell[CellGroupData[{

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Cell["Two ordinary circles and a thick one. ", "ExampleText",
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Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell[BoxData[
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Cell["TimeUsed", "ObjectName",
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If you perform a lengthy computation the total amount of CPU time \
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Cell[BoxData[
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Cell[TextData[{
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zone ",
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Cell["Timing", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["Timing", "MR"],
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\[FilledSmallSquare] ",
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Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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Cell[BoxData[
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Cell["\<\
Doing the computation again is considerably faster because tables \
were constructed the first time around. \
\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["ToBoxes", "ObjectName",
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  ButtonBox["MakeBoxes",
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Cell["ToCharacterCode", "ObjectName",
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Cell[TextData[{
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\[FilledSmallSquare] For other characters, they follow the Unicode standard. \
\n\[FilledSmallSquare] ",
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  Cell[BoxData[
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  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
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  ButtonBox["DigitQ",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell["\<\
This converts the string to its ASCII character code equivalents \
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\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[1]= ",
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[2]:= ",
  CellTags->"ToCharacterCode"],

Cell[BoxData[
    \("Everything is an expression."\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"ToCharacterCode"]
}, Open  ]],

Cell["ToDate", "ObjectName",
  CellTags->{"S0.1532", "ToDate"}],

Cell[TextData[{
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  StyleBox["ToDate[", "MR"],
  StyleBox["time", "TI"],
  StyleBox["]", "MR"],
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1900 to a date of the form ",
  Cell[BoxData[
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  StyleBox["y", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["m", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["d", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["h", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["m", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["s", "TI"],
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  CellTags->{"S0.1532", "ToDate"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["ToDate", "MR"],
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  StyleBox["AbsoluteTime", "MR"],
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  StyleBox["ToDate", "MR"],
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same time zone. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.3",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["FromDate",
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  ". \n\[FilledSmallSquare] Related package: ",
  ButtonBox["Miscellaneous`Calendar`",
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    ButtonStyle->"AddOnsLink"],
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell["\<\
September 10, 1931 was the date one billion seconds after January \
1, 1900. \
\>", "ExampleText",
  CellTags->"ToDate"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"ToDate"],

Cell[BoxData[
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  CellTags->"ToDate"]
}, Open  ]],

Cell["ToExpression", "ObjectName",
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Cell[TextData[{
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  StyleBox["ToExpression[", "MR"],
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  StyleBox["]", "MR"],
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  StyleBox["ToExpression[", "MR"],
  StyleBox["input", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["form", "TI"],
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\[FilledSmallSquare] ",
  StyleBox["ToExpression[", "MR"],
  StyleBox["input", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["form", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["h", "TI"],
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  CellTags->{"S0.1532", "ToExpression"}],

Cell[TextData[{
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  Cell[BoxData[
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  StyleBox["ToExpression[", "MR"],
  StyleBox["boxes", "TI"],
  StyleBox["]", "MR"],
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  StyleBox["ToExpression", "MR"],
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  StyleBox["ToExpression", "MR"],
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  StyleBox["ToExpression[", "MR"],
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  StyleBox["form", "TI"],
  StyleBox[",", "MR"],
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  ButtonBox["ToString",
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Cell["ToFileName", "ObjectName",
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Cell[TextData[{
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\[FilledSmallSquare] ",
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\\\"TI\\\"]\\)\"\>", "1"], TraditionalForm]], "InlineFormula"],
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  Cell[BoxData[
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        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"dir\\\\\\\"\\\", \
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  StyleBox[",", "MR"],
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\[FilledSmallSquare] ",
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  Cell[BoxData[
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Cell[TextData[{
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",
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Cell["ToRadicals", "ObjectName",
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Cell["This generates a string from the (evaluated) expression. ", \
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["TotalWidth", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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Cell["ToUpperCase", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell["Making ASCII from ascii. ", "ExampleText",
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["Tr", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["[[", "MR"],
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Cell[CellGroupData[{

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Cell[TextData[{
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Cell[BoxData[
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Cell[TextData[{
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  " and ",
  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  " and ",
  Cell[BoxData[
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  ". The quotient of the complex plane by this lattice is equivalent, as a \
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  Cell[BoxData[
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  ". \n",
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  " returns ",
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Cell[BoxData[
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Cell["\<\
The Weierstrass invariants depend only on the lattice, not on the \
particular generators. \
\>", "ExampleText",
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["WeierstrassP", "ObjectName",
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Cell[TextData[{
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  Cell[BoxData[
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  " gives the Weierstrass elliptic function ",
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Cell[TextData[{
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  " gives the value of ",
  Cell[BoxData[
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  " for which ",
  Cell[BoxData[
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  ". \n\[FilledSmallSquare] See Section\[NonBreakingSpace]",
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\[FilledSmallSquare] See ",
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Cell[TextData[{
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  " takes as a second argument the coefficients ",
  Cell[BoxData[
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  " of the equation of the elliptic curve under consideration (in Weierstrass \
normal form). If instead we know the periods of the curve, we start by using \
",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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  " has periods ",
  Cell[BoxData[
      \(TraditionalForm\`2  \[Omega]\)], "InlineFormula"],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`2  \[Omega]\^\[Prime]\)], "InlineFormula"],
  " (here ",
  Cell[BoxData[
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  " and ",
  Cell[BoxData[
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Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell[BoxData[
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    ContourPlot[
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Cell[TextData[{
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Cell["WeierstrassPPrime", "ObjectName",
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Cell[TextData[{
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  " gives the derivative of the Weierstrass elliptic function ",
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Cell[TextData[{
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  ". \n\[FilledSmallSquare] See Section\[NonBreakingSpace]",
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\[FilledSmallSquare] See ",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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  " takes as a second argument the coefficients ",
  Cell[BoxData[
      \(TraditionalForm\`{g\_2, g\_3}\)], "InlineFormula"],
  " of the equation of the elliptic curve under consideration (in Weierstrass \
normal form). If instead we know the periods of the curve, we start by using \
",
  StyleBox["WeierstrassInvariants", "MR"],
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"WeierstrassPPrime"],

Cell[BoxData[
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Cell[TextData[{
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  Cell[BoxData[
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  " and ",
  Cell[BoxData[
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    "InlineFormula"],
  " for some complex number ",
  Cell[BoxData[
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  ", the pair ",
  Cell[BoxData[
      \(TraditionalForm\`{p, q}\)], "InlineFormula"],
  " satisfies the defining equation for the curve, ",
  Cell[BoxData[
      \(TraditionalForm\`q\^2 = 4  p\^3 - \(g\_2\) p - g\_3\)], 
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  ". "
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Cell[CellGroupData[{

Cell[BoxData[
    \(p = WeierstrassP[u, {g\_2, g\_3}]; 
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      u -> Random[Real, {0, 1}, 50]\)], "Input",
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  CellTags->"WeierstrassPPrime"],

Cell[BoxData[
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  CellLabel->"Out[2]= ",
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Cell[TextData[{
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  Cell[BoxData[
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  " is an even elliptic function with poles only at the origin (in the \
fundamental parallelogram), so it must be expressible in terms of ",
  Cell[BoxData[
      \(TraditionalForm\`\[WeierstrassP]\)], "InlineFormula"],
  ". "
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[TextData[{
  "This plot illustrates that ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[WeierstrassP]\^\[Prime]\) \((g\_2, g\_3)\)\)], 
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  " has periods ",
  Cell[BoxData[
      \(TraditionalForm\`2  \[Omega]\)], "InlineFormula"],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`2  \[Omega]\^\[Prime]\)], "InlineFormula"],
  " (here ",
  Cell[BoxData[
      \(TraditionalForm\`2\)], "InlineFormula"],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`4  i\)], "InlineFormula"],
  "). "
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  CellTags->"WeierstrassPPrime"],

Cell["Evaluate the cell to see the graphic. ", "Text",
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Cell[BoxData[
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    ContourPlot[
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Cell["WeierstrassSigma", "ObjectName",
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Cell[TextData[{
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  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"g\\\\\\\"\\\", \\\"TI\
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  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"g\\\\\\\"\\\", \\\"TI\
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  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]", "MR"],
  " gives the Weierstrass sigma function ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma] \((u; g\_2, g\_3)\)\)], "InlineFormula"],
  ". "
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  CellTags->{"S0.1564", "WeierstrassSigma"}],

Cell[TextData[{
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    ButtonStyle->"MainBookLink"],
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  " by the differential equation ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[Sigma]\^\[Prime]\) \((z; g\_2, 
          g\_3)\)/\[Sigma] \((z; g\_2, g\_3)\) = \[Zeta] \((z; g\_2, 
          g\_3)\)\)], "InlineFormula"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["WeierstrassSigma", "MR"],
  " is not periodic and is therefore not strictly an elliptic function. \n\
\[FilledSmallSquare] See Section\[NonBreakingSpace]",
  ButtonBox["3.2.11",
    ButtonData:>{"3.2.11", "11.3"},
    ButtonStyle->"MainBookLink"],
  " for a discussion of argument conventions for elliptic and related \
functions. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.2.11",
    ButtonData:>{"3.2.11", "11.12"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"WeierstrassSigma"],

Cell[TextData[{
  StyleBox["WeierstrassSigma", "MR"],
  " takes as a second argument the coefficients ",
  Cell[BoxData[
      \(TraditionalForm\`{g\_2, g\_3}\)], "InlineFormula"],
  " of the equation of the elliptic curve under consideration (in Weierstrass \
normal form). If instead we know the periods of the curve, we start by using \
",
  StyleBox["WeierstrassInvariants", "MR"],
  ". (We need bignums to get the next example to work.) "
}], "ExampleText",
  CellTags->"WeierstrassSigma"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{
      RowBox[{
        RowBox[{"{", 
          RowBox[{"\[Omega]", ",", 
            SuperscriptBox["\[Omega]", "\[Prime]",
              MultilineFunction->None]}], "}"}], 
        "=", \({1, N[2, 50]\ \[ImaginaryI]}\)}], ";", 
      RowBox[{\({g\_2, g\_3}\), "=", 
        RowBox[{"WeierstrassInvariants", "[", 
          RowBox[{"{", 
            RowBox[{"\[Omega]", ",", 
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  CellTags->"WeierstrassSigma"],

Cell[BoxData[
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1355}\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"WeierstrassSigma"]
}, Open  ]],

Cell[TextData[{
  "The Weierstrass sigma function satisfies the following \
quasi\[Hyphen]periodicity condition: if ",
  Cell[BoxData[
      \(TraditionalForm\`v\)], "InlineFormula"],
  " is any element of the lattice ",
  Cell[BoxData[
      \(TraditionalForm\`L\)], "InlineFormula"],
  " generated by the periods ",
  Cell[BoxData[
      \(TraditionalForm\`2  \[Omega]\)], "InlineFormula"],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`2  \[Omega]\^\[Prime]\)], "InlineFormula"],
  ", then ",
  Cell[BoxData[
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            v)\) = \(e\^\(a  u + b\)\) \[Sigma] \((u)\)\)], "InlineFormula"],
  " for all ",
  Cell[BoxData[
      \(TraditionalForm\`u\)], "InlineFormula"],
  ", where ",
  Cell[BoxData[
      \(TraditionalForm\`a, b\)], "InlineFormula"],
  " are constants depending on ",
  Cell[BoxData[
      \(TraditionalForm\`v\)], "InlineFormula"],
  ". For ",
  Cell[BoxData[
      \(TraditionalForm\`v \[NotElement] 2  L\)], "InlineFormula"],
  ", we have ",
  Cell[BoxData[
      \(TraditionalForm\`a = 2  \[Zeta] \((v/2)\)\)], "InlineFormula"],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`b = a  v/2 + \[Pi]  i\)], "InlineFormula"],
  ", where ",
  Cell[BoxData[
      \(TraditionalForm\`\[Zeta]\)], "InlineFormula"],
  " is the Weierstrass zeta function. Here is an example, with ",
  Cell[BoxData[
      \(TraditionalForm\`v = 2  \[Omega]\)], "InlineFormula"],
  ". "
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Cell[CellGroupData[{

Cell[BoxData[
    \(a = 2\ WeierstrassZeta[\[Omega], {g\_2, g\_3}]\)], "Input",
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  CellTags->"WeierstrassSigma"],

Cell[BoxData[
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  CellTags->"WeierstrassSigma"]
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Cell[CellGroupData[{

Cell[BoxData[
    \(b = a  \[Omega] + \[ImaginaryI]  \[Pi]\)], "Input",
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  CellTags->"WeierstrassSigma"],

Cell[BoxData[
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  CellLabel->"Out[3]= ",
  CellTags->"WeierstrassSigma"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"WeierstrassSigma"],

Cell[BoxData[
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  CellLabel->"Out[4]= ",
  CellTags->"WeierstrassSigma"]
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Cell[BoxData[
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  CellTags->"WeierstrassSigma"],

Cell["WeierstrassZeta", "ObjectName",
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Cell[TextData[{
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  StyleBox["WeierstrassZeta[", "MR"],
  StyleBox["u", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"g\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"g\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "3"], TraditionalForm]], "InlineFormula"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox["]", "MR"],
  " gives the Weierstrass zeta function ",
  Cell[BoxData[
      \(TraditionalForm\`\[Zeta] \((u; g\_2, g\_3)\)\)], "InlineFormula"],
  ". "
}], "Usage",
  CellTags->{"S0.1564", "WeierstrassZeta"}],

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\[FilledSmallSquare] See Section\[NonBreakingSpace]",
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\[FilledSmallSquare] See ",
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  StyleBox["PageWidth", "MR"],
  " option effectively set to ",
  StyleBox["Infinity", "MR"],
  ", does not give ",
  StyleBox["In", "MR"],
  " and ",
  StyleBox["Out", "MR"],
  " labels, and does not give any banner when it starts up. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.5"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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  ", ",
  ButtonBox["$Linked",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$CommandLine",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1577", "$BatchOutput"}],

Cell["$ByteOrdering", "ObjectName",
  CellTags->{"S0.1577", "$ByteOrdering"}],

Cell[TextData[{
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system. "
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  CellTags->{"S0.1577", "$ByteOrdering"}],

Cell[TextData[{
  "\[FilledSmallSquare] Possible values of ",
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  StyleBox["+1", "MR"],
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  StyleBox["+1", "MR"],
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processors); ",
  StyleBox["-1", "MR"],
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\[FilledSmallSquare] ",
  StyleBox["+1", "MR"],
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  StyleBox["+1", "MR"],
  " is the order obtained from ",
  StyleBox["IntegerDigits[", "MR"],
  StyleBox["n", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["256]", "MR"],
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  StyleBox["$ByteOrdering", "MR"],
  " gives the default setting for the ",
  StyleBox["ByteOrdering", "MR"],
  " option in ",
  StyleBox["Import", "MR"],
  " and ",
  StyleBox["Export", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.13"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$ProcessorType",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1577", "$ByteOrdering"}],

Cell["$CharacterEncoding", "ObjectName",
  CellTags->{"S0.1577", "$CharacterEncoding"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$CharacterEncoding", "MR"],
  " specifies the default raw character encoding to use for input and output \
functions. "
}], "Usage",
  CellTags->{"S0.1577", "$CharacterEncoding"}],

Cell[TextData[{
  "\[FilledSmallSquare] The default setting for ",
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  StyleBox["$SystemCharacterEncoding", "MR"],
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  StyleBox["$CharacterEncoding", "MR"],
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  StyleBox["None", "MR"],
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  StyleBox["[", "MR"],
  StyleBox["Name", "TI"],
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  StyleBox[":", "MR"],
  StyleBox["xxxx", "TI"],
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}], "Notes",
  CellTags->{"S0.1577", "$CharacterEncoding"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ASCII\\\\\\\\\\\
\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"full ASCII, with control \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"EUC\\\\\\\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ISOLatin1\\\\\\\
\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"ISO 8859\[Hyphen]1 \
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ISOLatin2\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ISOLatin3\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ISOLatin4\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"MacintoshRoman\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"PrintableASCII\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ShiftJIS\\\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"Symbol\\\\\\\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"WindowsANSI\\\\\
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"ZapfDingbats\\\\\
\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\>", "\<\"Zapf dingbats font \
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Cell[TextData[{
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  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["\"", "MR"],
  StyleBox["encoding", "TI"],
  StyleBox["\"", "MR"],
  " characters that are included in the encoding can be input in their raw 8\
\[Hyphen] or 16\[Hyphen]bit form, and will be output in this form. \n\
\[FilledSmallSquare] Unencoded characters can be input and will be output in \
standard \[Backslash]",
  StyleBox["[", "MR"],
  StyleBox["Name", "TI"],
  StyleBox["]", "MR"],
  " or \[Backslash]",
  StyleBox[":", "MR"],
  StyleBox["xxxx", "TI"],
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resetting the value of ",
  StyleBox["$CharacterEncoding", "MR"],
  " has an immediate effect on standard input and output in a ",
  StyleBox["Mathematica", "TI"],
  " session. \n\[FilledSmallSquare] When using the notebook front end, raw \
character encodings are normally handled automatically based on the fonts you \
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  " connection to the kernel. \n\[FilledSmallSquare] ",
  StyleBox["$CharacterEncoding", "MR"],
  " can be set to a list of the form ",
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  StyleBox["class", "TI"],
  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
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  StyleBox["\"", "MR"],
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  Cell[BoxData[
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  StyleBox[",", "MR"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"n\\\\\\\"\\\", \\\"TI\
\\\"]\\)\"\>", "2"], TraditionalForm]], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["\"", "MR"],
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"c\\\\\\\"\\\", \\\"TI\
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  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[",", "MR"],
  " \[Ellipsis] ",
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  ". The ",
  StyleBox["class", "TI"],
  " defines the general form of encoding; the ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"n\\\\\\\"\\\", \\\"TI\
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      FormBox[
        SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"c\\\\\\\"\\\", \\\"TI\
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  StyleBox["class", "TI"],
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}], "Notes",
  CellTags->{"S0.1577", "$CharacterEncoding"}],

Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"8Bit\\\\\\\\\\\\\
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  CellTags->"$CharacterEncoding"],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$CharacterEncoding", "MR"],
  " affects the input and output of all characters, including those in symbol \
names and comments. \n\[FilledSmallSquare] ",
  StyleBox["$CharacterEncoding", "MR"],
  " also affects characters that appear in ",
  StyleBox["Text", "MR"],
  " graphics primitives. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.7.8",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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  ButtonBox["FromCharacterCode",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ToCharacterCode",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$SystemCharacterEncoding",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$ByteOrdering",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
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Cell["$CommandLine", "ObjectName",
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Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$CommandLine", "MR"],
  " is a list of strings giving the elements of the original operating system \
command line with which ",
  StyleBox["Mathematica", "TI"],
  " was invoked. "
}], "Usage",
  CellTags->{"S0.1578", "$CommandLine"}],

Cell[TextData[{
  "\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.7"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Environment",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$LaunchDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$BatchInput",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$BatchOutput",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Linked",
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  ", ",
  ButtonBox["In",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1578", "$CommandLine"}],

Cell["$Context", "ObjectName",
  CellTags->{"S0.1579", "$Context"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$Context", "MR"],
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}], "Usage",
  CellTags->{"S0.1579", "$Context"}],

Cell[TextData[{
  "\[FilledSmallSquare] Contexts are specified by strings of the form ",
  StyleBox["\"", "MR"],
  StyleBox["name", "TI"],
  StyleBox["`\"", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$Context", "MR"],
  " is modified by ",
  StyleBox["Begin", "MR"],
  ", ",
  StyleBox["BeginPackage", "MR"],
  ", ",
  StyleBox["End", "MR"],
  " and ",
  StyleBox["EndPackage", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$Context", "MR"],
  " is a rough analog for ",
  StyleBox["Mathematica", "TI"],
  " symbols of the current working directory for files in many operating \
systems. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.6.8",
    ButtonData:>{"2.6.8", "8.7"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Context",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$ContextPath",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1579", "$Context"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"$Context"],

Cell[TextData[{
  " Every symbol (function or variable) in ",
  StyleBox["Mathematica", "TI"],
  " is associated with a ",
  StyleBox["context", "TI"],
  ". This allows the same name to be used for different objects, if they \
belong to different contexts. The sequence of commands below will demonstrate \
this. "
}], "ExampleText",
  CellTags->"$Context"],

Cell[TextData[{
  "There is a notion of a current context for symbol lookup. Here is the \
current context when you start ",
  StyleBox["Mathematica", "TI"],
  ". "
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \($Context\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Global`"\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"$Context"]
}, Open  ]],

Cell["\<\
The following name does not exist in memory yet. When you refer to \
it, a new symbol is created and is assigned to the current context. \
\>", \
"ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(xxxx\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \(xxxx\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"$Context"]
}, Open  ]],

Cell["This is the context to which the new symbol belongs. ", "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Context[xxxx]\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Global`"\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"$Context"]
}, Open  ]],

Cell["This is its full name. ", "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?xxxx\)\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Global`xxxx"\)], "Print",
  CellTags->"$Context"]
}, Open  ]],

Cell[TextData[{
  "Assigning a new value to ",
  StyleBox["$Context", "MR"],
  " changes the current context. Notice that the context name is a string \
ending with a backquote. "
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \($Context = "\<Temporary1`\>"\)], "Input",
  CellLabel->"In[5]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Temporary1`"\)], "Output",
  CellLabel->"Out[5]= ",
  CellTags->"$Context"]
}, Open  ]],

Cell["\<\
When you refer to a name that has not been used before, once more a \
new symbol is created and assigned to the current context. \
\>", \
"ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(xxyy\)], "Input",
  CellLabel->"In[6]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \(xxyy\)], "Output",
  CellLabel->"Out[6]= ",
  CellTags->"$Context"]
}, Open  ]],

Cell["This is its full name. ", "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?xxyy\)\)], "Input",
  CellLabel->"In[7]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Temporary1`xxyy"\)], "Print",
  CellTags->"$Context"]
}, Open  ]],

Cell["We change the context yet again. ", "ExampleText",
  CellTags->"$Context"],

Cell[BoxData[
    \($Context = "\<Temporary2`\>"; \)], "Input",
  CellLabel->"In[8]:= ",
  CellTags->"$Context"],

Cell[TextData[{
  "The symbol we created in the ",
  StyleBox["Global`", "MR"],
  " context is still accessible under its short name, because ",
  StyleBox["Global`", "MR"],
  " is in the context search path, as explained in the Further Examples for \
",
  ButtonBox["$ContextPath",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?xxxx\)\)], "Input",
  CellLabel->"In[9]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Global`xxxx"\)], "Print",
  CellTags->"$Context"]
}, Open  ]],

Cell[TextData[{
  "The symbol we created in the ",
  StyleBox["Temporary1`", "MR"],
  " context is not accessible under its short name, because ",
  StyleBox["Temporary1`", "MR"],
  " is not in the context search path. "
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?xxyy\)\)], "Input",
  CellLabel->"In[10]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \(Information::"notfound" \(\(:\)\(\ \)\) 
      "Symbol \!\(\"xxyy\"\) not found."\)], "Message",
  CellTags->"$Context"]
}, Open  ]],

Cell[TextData[{
  "Although ",
  StyleBox["xxyy", "MR"],
  " is no longer recognized, the symbol is still accessible under its full \
name. "
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?Temporary1`xxyy\)\)], "Input",
  CellLabel->"In[11]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Temporary1`xxyy"\)], "Print",
  CellTags->"$Context"]
}, Open  ]],

Cell[TextData[{
  "A reference to ",
  StyleBox["xxxx", "MR"],
  " accesses ",
  StyleBox["Global`xxxx", "MR"],
  ", while a reference to ",
  StyleBox["xxyy", "MR"],
  " creates a new symbol in the current context. "
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \({xxxx, xxyy}; \n\(?xxyy\)\)], "Input",
  CellLabel->"In[12]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Temporary2`xxyy"\)], "Print",
  CellTags->"$Context"]
}, Open  ]],

Cell[TextData[{
  "This shows all the symbols currently accessible under short names that \
start with ",
  StyleBox["xx", "MR"],
  ". "
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?xx*\)\)], "Input",
  CellLabel->"In[13]:= ",
  CellTags->"$Context"],

Cell["xxxx xxyy", "Print",
  CellTags->"$Context"]
}, Open  ]],

Cell[TextData[{
  "This shows all symbols (in all contexts) whose short names start with ",
  StyleBox["xx", "MR"],
  ". Note that ",
  StyleBox["Temporary1`xxyy", "MR"],
  " and ",
  StyleBox["xxyy", "MR"],
  " (which currently stands for ",
  StyleBox["Temporary2`xxyy", "MR"],
  ") are completely independent symbols, and can be given different values \
and rules. When you use the short name ",
  StyleBox["xxyy", "MR"],
  " you are referring to one or the other, depending on the current context. \
"
}], "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?*`xx*\)\)], "Input",
  CellLabel->"In[14]:= ",
  CellTags->"$Context"],

Cell["\<\
DSolve`DSolve2x2Dump`xx
Integrate`ImproperDump`xx
Integrate`NLtheoremDump`xx
System`FunctionInterpolationDump`xx
System`ISFDump`xx
DSolve`DSolveLinDump`xx1
DSolve`DSolveLinDump`xx1$
DSolve`DSolveLinDump`xx2
DSolve`DSolveLinDump`xx2$
DSolve`DSolveNL1Dump`xxx
xxxx
DSolve`DSolveNL1Dump`xxx$
Temporary1`xxyy
xxyy
DSolve`DSolve2x2Dump`xx$
Integrate`NLtheoremDump`xx$
System`FunctionInterpolationDump`xx$\
\>", "Print",
  CellTags->"$Context"]
}, Open  ]],

Cell["We revert to the default context. ", "ExampleText",
  CellTags->"$Context"],

Cell[CellGroupData[{

Cell[BoxData[
    \($Context = "\<Global`\>"\)], "Input",
  CellLabel->"In[15]:= ",
  CellTags->"$Context"],

Cell[BoxData[
    \("Global`"\)], "Output",
  CellLabel->"Out[15]= ",
  CellTags->"$Context"]
}, Open  ]],

Cell[TextData[{
  " Usually the current context is changed not by means of an assignment to \
",
  StyleBox["$Context", "MR"],
  ", but rather indirectly, with the ",
  StyleBox["Begin", "MR"],
  " command. This is so that the old context can be restored more easily. See \
the Further Examples for ",
  ButtonBox["Begin",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "ExampleText",
  CellTags->"$Context"],

Cell["$ContextPath", "ObjectName",
  CellTags->{"S0.1580", "$ContextPath"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$ContextPath", "MR"],
  " is a global variable that gives a list of contexts, after ",
  StyleBox["$Context", "MR"],
  ", to search in trying to find a symbol that has been entered. "
}], "Usage",
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Cell[TextData[{
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  StyleBox["`\"", "MR"],
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  StyleBox["$ContextPath", "MR"],
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  StyleBox["Begin", "MR"],
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  StyleBox["BeginPackage", "MR"],
  ", ",
  StyleBox["End", "MR"],
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  StyleBox["EndPackage", "MR"],
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  StyleBox["$ContextPath", "MR"],
  " is a rough analog for ",
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.6.8",
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}], "Notes",
  CellTags->{"S0.1580", "$ContextPath"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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path, which is stored in ",
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[1]= ",
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Cell[TextData[{
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no longer accessed under ",
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Cell[CellGroupData[{

Cell["$ContextPath={\"Global`\"};  Log[1]", "Input",
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Cell[BoxData[
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  CellLabel->"Out[2]= ",
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Cell["\<\
It is still accessible under the full name, like any other symbol. \
\
\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[3]= ",
  CellTags->"$ContextPath"]
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Cell["Here we restore the context path to its initial value. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[4]:= ",
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Cell[BoxData[
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  CellLabel->"Out[4]= ",
  CellTags->"$ContextPath"]
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Cell[TextData[{
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to ",
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Further Examples for ",
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Cell["$CreationDate", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["month", "TI"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["day", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["hour", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["minute", "TI"],
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  " ",
  StyleBox["second", "TI"],
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  StyleBox["Date", "MR"],
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$ReleaseNumber",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FileDate",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$InstallationDate",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1581", "$CreationDate"}],

Cell["$CurrentLink", "ObjectName",
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Cell[TextData[{
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  StyleBox["MathLink", "TI"],
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Cell[TextData[{
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  StyleBox[":Arguments:", "MR"],
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  StyleBox["Install", "MR"],
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.9",
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  ButtonBox["Links",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Input",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1581", "$CurrentLink"}],

Cell["$Display", "ObjectName",
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Cell[TextData[{
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  CellTags->{"S0.1582", "$Display"}],

Cell[TextData[{
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  StyleBox["The Mathematica Book: ", "TI"],
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    ButtonStyle->"MainBookLink"],
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  ButtonBox["Section 2.13.1",
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    ButtonStyle->"MainBookLink"],
  "."
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  CellTags->{"S0.1582", "$Display"}],

Cell["$DisplayFunction", "ObjectName",
  CellTags->{"S0.1583", "$DisplayFunction"}],

Cell[TextData[{
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  CellTags->{"S0.1583", "$DisplayFunction"}],

Cell[TextData[{
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  StyleBox["#]&", "MR"],
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",
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display. \n\[FilledSmallSquare] See ",
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  ButtonBox["Section 2.9.1",
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    ButtonStyle->"MainBookLink"],
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    ButtonStyle->"RefGuideLink"],
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  ButtonBox["Put",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Run",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$SoundDisplayFunction",
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}], "Notes",
  CellTags->{"S0.1583", "$DisplayFunction"}]
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[CellGroupData[{

Cell[TextData[{
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Cell["$Echo", "ObjectName",
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Cell[TextData[{
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}], "Usage",
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Cell[TextData[{
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
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    ButtonStyle->"MainBookLink"],
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}], "Notes",
  CellTags->{"S0.1584", "$Echo"}],

Cell["$Epilog", "ObjectName",
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Cell[TextData[{
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}], "Usage",
  CellTags->{"S0.1585", "$Epilog"}],

Cell[TextData[{
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
    ButtonData:>{"2.13.1", "1.35"},
    ButtonStyle->"MainBookLink"],
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  ", ",
  ButtonBox["Quit",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Dialog",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1585", "$Epilog"}],

Cell["$ExportFormats", "ObjectName",
  CellTags->{"S0.1585", "$ExportFormats"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$ExportFormats", "MR"],
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  StyleBox["Mathematica", "TI"],
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}], "Usage",
  CellTags->{"S0.1585", "$ExportFormats"}],

Cell[TextData[{
  "\[FilledSmallSquare] The strings that appear in ",
  StyleBox["$ExportFormats", "MR"],
  " are the possible third arguments to ",
  StyleBox["Export", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.11.3",
    ButtonData:>{"1.11.3", "3.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$ImportFormats",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Export",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Packages",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1585", "$ExportFormats"}],

Cell["$Failed", "ObjectName",
  CellTags->{"S0.1586", "$Failed"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$Failed", "MR"],
  " is a special symbol returned by certain functions when they cannot do \
what they were asked to do. "
}], "Usage",
  CellTags->{"S0.1586", "$Failed"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["Get", "MR"],
  " returns ",
  StyleBox["$Failed", "MR"],
  " when it cannot find the file or other object that was specified. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.1",
    ButtonData:>{"2.11.1", "1.6"},
    ButtonStyle->"MainBookLink"],
  "."
}], "Notes",
  CellTags->{"S0.1586", "$Failed"}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"$Failed"],

Cell[TextData[{
  "The file ",
  StyleBox["missing", "MR"],
  " does not exist in the current path, and so ",
  StyleBox["Get", "MR"],
  " fails to find it. "
}], "ExampleText",
  CellTags->"$Failed"],

Cell[CellGroupData[{

Cell[BoxData[
    \(<< missing\)], "Input",
  CellLabel->"In[1]:= ",
  CellTags->"$Failed"],

Cell[BoxData[
    \(Get::"noopen" \(\(:\)\(\ \)\) 
      "Cannot open \!\(\"missing\"\)."\)], "Message",
  CellTags->"$Failed"],

Cell[BoxData[
    \($Failed\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"$Failed"]
}, Open  ]],

Cell["$FormatType", "ObjectName",
  CellTags->{"S0.1586", "$FormatType"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$FormatType", "MR"],
  " gives the default format type to use for text that appears in graphics. "
}], "Usage",
  CellTags->{"S0.1586", "$FormatType"}],

Cell[TextData[{
  "\[FilledSmallSquare] The default setting for the standard notebook front \
end is ",
  StyleBox["$FormatType", "MR"],
  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["StandardForm", "MR"],
  ". \n\[FilledSmallSquare] The default setting for the text\[Hyphen]based \
front end is ",
  StyleBox["$FormatType", "MR"],
  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["OutputForm", "MR"],
  ". \n\[FilledSmallSquare] A common alternative setting is ",
  StyleBox["TraditionalForm", "MR"],
  ". \n\[FilledSmallSquare] Box\[Hyphen]based format types such as ",
  StyleBox["StandardForm", "MR"],
  " and ",
  StyleBox["TraditionalForm", "MR"],
  " can be used only when a notebook front end is present. \n\
\[FilledSmallSquare] ",
  StyleBox["$FormatType", "MR"],
  " gives the default value for the ",
  StyleBox["FormatType", "MR"],
  " option in graphics. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.9.15",
    ButtonData:>{"2.9.15", "15.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$TextStyle",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["FormatType",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1586", "$FormatType"}],

Cell["$FrontEnd", "ObjectName",
  CellTags->{"S0.1586", "$FrontEnd"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$FrontEnd", "MR"],
  " is a global variable that specifies to what front end object, if any, the \
kernel is currently connected. "
}], "Usage",
  CellTags->{"S0.1586", "$FrontEnd"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$FrontEnd", "MR"],
  " is either a ",
  StyleBox["FrontEndObject", "MR"],
  ", or ",
  StyleBox["Null", "MR"],
  ". \n\[FilledSmallSquare] You can use ",
  StyleBox["Options", "MR"],
  " and ",
  StyleBox["SetOptions", "MR"],
  " on ",
  StyleBox["$FrontEnd", "MR"],
  " to read and set global options for the front end. \n\[FilledSmallSquare] \
See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.10.4",
    ButtonData:>{"2.10.4", "4.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$Notebooks",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1586", "$FrontEnd"}],

Cell["$HistoryLength", "ObjectName",
  CellTags->{"S0.1586", "$HistoryLength"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$HistoryLength", "MR"],
  " specifies the number of previous lines of input and output to keep in a \
",
  StyleBox["Mathematica", "TI"],
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}], "Usage",
  CellTags->{"S0.1586", "$HistoryLength"}],

Cell[TextData[{
  "\[FilledSmallSquare] The default setting for ",
  StyleBox["$HistoryLength", "MR"],
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  StyleBox["Infinity", "MR"],
  ". \n\[FilledSmallSquare] Values of ",
  StyleBox["In[", "MR"],
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
  " and ",
  StyleBox["Out[", "MR"],
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
  " corresponding to lines before those kept are explicitly cleared. \n\
\[FilledSmallSquare] Using smaller values of ",
  StyleBox["$HistoryLength", "MR"],
  " can save substantial amounts of memory in a ",
  StyleBox["Mathematica", "TI"],
  " session. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$Line",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1586", "$HistoryLength"}],

Cell["$HomeDirectory", "ObjectName",
  CellTags->{"S0.1586", "$HomeDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$HomeDirectory", "MR"],
  " gives your \[OpenCurlyDoubleQuote]home\[CloseCurlyDoubleQuote] directory. \
"
}], "Usage",
  CellTags->{"S0.1586", "$HomeDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
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  " returns the full name of the directory as a string. \n\
\[FilledSmallSquare] On multi\[Hyphen]user operating systems, ",
  StyleBox["$HomeDirectory", "MR"],
  " gives the main directory for the current user. \n\[FilledSmallSquare] See \
",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.4",
    ButtonData:>{"2.11.4", "4.9"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Directory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ParentDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$UserName",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$LaunchDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$TopDirectory",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1586", "$HomeDirectory"}],

Cell["$IgnoreEOF", "ObjectName",
  CellTags->{"S0.1587", "$IgnoreEOF"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$IgnoreEOF", "MR"],
  " specifies whether ",
  StyleBox["Mathematica", "TI"],
  " should terminate when it receives an end\[Hyphen]of\[Hyphen]file \
character as input. "
}], "Usage",
  CellTags->{"S0.1587", "$IgnoreEOF"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$IgnoreEOF", "MR"],
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  StyleBox["False", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$IgnoreEOF", "MR"],
  " is assumed to be ",
  StyleBox["False", "MR"],
  " if the input to ",
  StyleBox["Mathematica", "TI"],
  " comes from a file, rather than an interactive device. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
    ButtonData:>{"2.13.1", "1.37"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section A.7.5",
    ButtonData:>{"A.7.5", "5.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Exit",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Quit",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$BatchInput",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1587", "$IgnoreEOF"}],

Cell["$ImportFormats", "ObjectName",
  CellTags->{"S0.1587", "$ImportFormats"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$ImportFormats", "MR"],
  " gives a list of import formats currently supported in your ",
  StyleBox["Mathematica", "TI"],
  " system. "
}], "Usage",
  CellTags->{"S0.1587", "$ImportFormats"}],

Cell[TextData[{
  "\[FilledSmallSquare] The strings that appear in ",
  StyleBox["$ImportFormats", "MR"],
  " are the possible second arguments to ",
  StyleBox["Import", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 1.11.3",
    ButtonData:>{"1.11.3", "3.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$ExportFormats",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Import",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Packages",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1587", "$ImportFormats"}],

Cell["$InitialDirectory", "ObjectName",
  CellTags->{"S0.1587", "$InitialDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$InitialDirectory", "MR"],
  " gives the current directory at the time when the current ",
  StyleBox["Mathematica", "TI"],
  " session was started. "
}], "Usage",
  CellTags->{"S0.1587", "$InitialDirectory"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$InitialDirectory", "MR"],
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\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.4",
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    ButtonStyle->"MainBookLink"],
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    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$LaunchDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$CommandLine",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1587", "$InitialDirectory"}],

Cell["$InstallationDate", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["month", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["day", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["hour", "TI"],
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  StyleBox["minute", "TI"],
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}], "Notes",
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Cell["$Input", "ObjectName",
  CellTags->{"S0.1589", "$Input"}],

Cell[TextData[{
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}], "Usage",
  CellTags->{"S0.1589", "$Input"}],

Cell[TextData[{
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  ButtonBox["Streams",
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  ButtonBox["DirectoryName",
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  ButtonBox["$BatchInput",
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Cell["$Inspector", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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  StyleBox["Dialog[", "MR"],
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  StyleBox["]&", "MR"],
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Cell["$IterationLimit", "ObjectName",
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Cell[TextData[{
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Cell[TextData[{
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\[FilledSmallSquare] ",
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",
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}], "Notes",
  CellTags->{"S0.1591", "$IterationLimit"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[1]= ",
  CellTags->"$IterationLimit"]
}, Open  ]],

Cell[TextData[{
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  StyleBox["f", "MR"],
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}], "ExampleText",
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Cell[BoxData[
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  CellTags->"$IterationLimit"],

Cell["The definition can lead to infinite iteration. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["Add a termination rule. ", "ExampleText",
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Cell[BoxData[
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  CellLabel->"In[4]:= ",
  CellTags->"$IterationLimit"],

Cell["Now the calculation will finish. ", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[5]= ",
  CellTags->"$IterationLimit"]
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Cell[TextData[{
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exceeds the value of ",
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}], "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[6]= ",
  CellTags->"$IterationLimit"]
}, Open  ]],

Cell[TextData[{
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}], "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"Out[7]= ",
  CellTags->"$IterationLimit"]
}, Open  ]],

Cell["\<\
Now the calculation will finish within the allowed number of \
iterations. \
\>", "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[8]:= ",
  CellTags->"$IterationLimit"],

Cell[BoxData[
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  CellLabel->"Out[8]= ",
  CellTags->"$IterationLimit"]
}, Open  ]],

Cell[BoxData[
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  CellLabel->"In[9]:= ",
  CellTags->"$IterationLimit"],

Cell["$Language", "ObjectName",
  CellTags->{"S0.1592", "$Language"}],

Cell[TextData[{
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  StyleBox["$Language", "MR"],
  " is a list of strings which give the names of languages to use for \
messages. "
}], "Usage",
  CellTags->{"S0.1592", "$Language"}],

Cell[TextData[{
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  StyleBox["::", "MR"],
  StyleBox["tag", "TI"],
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  StyleBox["s", "TI"],
  StyleBox["::", "MR"],
  StyleBox["tag", "TI"],
  StyleBox["::", "MR"],
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  StyleBox["tag", "TI"],
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    ButtonStyle->"MainBookLink"],
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  ButtonBox["Section 2.13.1",
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    ButtonStyle->"MainBookLink"],
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    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["LanguageCategory",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
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Cell["$LaunchDirectory", "ObjectName",
  CellTags->{"S0.1592", "$LaunchDirectory"}],

Cell[TextData[{
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  StyleBox["$LaunchDirectory", "MR"],
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  StyleBox["Mathematica", "TI"],
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}], "Usage",
  CellTags->{"S0.1592", "$LaunchDirectory"}],

Cell[TextData[{
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\[FilledSmallSquare] ",
  StyleBox["$LaunchDirectory", "MR"],
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  StyleBox["Mathematica", "TI"],
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",
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.4",
    ButtonData:>{"2.11.4", "4.9"},
    ButtonStyle->"MainBookLink"],
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  ButtonBox["$TopDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$CommandLine",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$InitialDirectory",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$InstallationDate",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1592", "$LaunchDirectory"}],

Cell["$Line", "ObjectName",
  CellTags->{"S0.1593", "$Line"}],

Cell[TextData[{
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  StyleBox["$Line", "MR"],
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"
}], "Usage",
  CellTags->{"S0.1593", "$Line"}],

Cell[TextData[{
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  StyleBox["$Line", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
    ButtonData:>{"2.13.1", "1.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["In",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Out",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$HistoryLength",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1593", "$Line"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"$Line"],

Cell["This resets the numbering of input and output prompts. ", "ExampleText",
  CellTags->"$Line"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"$Line"],

Cell[BoxData[
    \(0\)], "Output",
  CellLabel->"Out[0]= ",
  CellTags->"$Line"]
}, Open  ]],

Cell["Here is some input. ", "ExampleText",
  CellTags->"$Line"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(20!\)\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"$Line"],

Cell[BoxData[
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  CellLabel->"Out[1]= ",
  CellTags->"$Line"]
}, Open  ]],

Cell[TextData[{
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  StyleBox["$Line", "MR"],
  " is now ",
  Cell[BoxData[
      \(TraditionalForm\`2\)], "InlineFormula"],
  ". "
}], "ExampleText",
  CellTags->"$Line"],

Cell[CellGroupData[{

Cell[BoxData[
    \($Line\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"$Line"],

Cell[BoxData[
    \(2\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"$Line"]
}, Open  ]],

Cell[TextData[{
  "Results from these evaluations are saved as values for ",
  StyleBox["Out", "MR"],
  ". "
}], "ExampleText",
  CellTags->"$Line"],

Cell[CellGroupData[{

Cell[BoxData[
    \({Out[1], Out[2]}\)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"$Line"],

Cell[BoxData[
    \({2432902008176640000, 2}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"$Line"]
}, Open  ]],

Cell[TextData[{
  "Resetting ",
  StyleBox["$Line", "MR"],
  " can be used for conserving memory, since this will cause the saved \
results to be overwritten by any new results. "
}], "ExampleText",
  CellTags->"$Line"],

Cell[CellGroupData[{

Cell[BoxData[
    \($Line = 0\)], "Input",
  CellLabel->"In[5]:= ",
  CellTags->"$Line"],

Cell[BoxData[
    \(0\)], "Output",
  CellLabel->"Out[0]= ",
  CellTags->"$Line"]
}, Open  ]],

Cell["Here is some new input. ", "ExampleText",
  CellTags->"$Line"],

Cell[CellGroupData[{

Cell[BoxData[
    \(2 + 5\)], "Input",
  CellLabel->"In[6]:= ",
  CellTags->"$Line"],

Cell[BoxData[
    \(7\)], "Output",
  CellLabel->"Out[1]= ",
  CellTags->"$Line"]
}, Open  ]],

Cell["The previous result was overwritten by the new result. ", "ExampleText",
  CellTags->"$Line"],

Cell[CellGroupData[{

Cell[BoxData[
    \({Out[1], Out[2]}\)], "Input",
  CellLabel->"In[7]:= ",
  CellTags->"$Line"],

Cell[BoxData[
    \({7, 2}\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"$Line"]
}, Open  ]],

Cell["$Linked", "ObjectName",
  CellTags->{"S0.1594", "$Linked"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$Linked", "MR"],
  " is ",
  StyleBox["True", "MR"],
  " if the ",
  StyleBox["Mathematica", "TI"],
  " kernel is being run through ",
  StyleBox["MathLink", "TI"],
  ". "
}], "Usage",
  CellTags->{"S0.1594", "$Linked"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$Linked", "MR"],
  " is ",
  StyleBox["True", "MR"],
  " when ",
  StyleBox["Mathematica", "TI"],
  " is being run with a front end. \n\[FilledSmallSquare] ",
  StyleBox["$Linked", "MR"],
  " is typically ",
  StyleBox["False", "MR"],
  " when ",
  StyleBox["Mathematica", "TI"],
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See also: ",
  ButtonBox["$CommandLine",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$BatchInput",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$BatchOutput",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Notebooks",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1594", "$Linked"}],

Cell["$MachineDomain", "ObjectName",
  CellTags->{"S0.1594", "$MachineDomain"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MachineDomain", "MR"],
  " is a string which gives the name of the network domain for the computer \
on which ",
  StyleBox["Mathematica", "TI"],
  " is being run, if such a name is defined. "
}], "Usage",
  CellTags->{"S0.1594", "$MachineDomain"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MachineDomain", "MR"],
  " is ",
  StyleBox["\"\"", "MR"],
  " if no name is defined. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.16"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$MachineName",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MachineID",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1594", "$MachineDomain"}],

Cell["$MachineEpsilon", "ObjectName",
  CellTags->{"S0.1595", "$MachineEpsilon"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MachineEpsilon", "MR"],
  " gives the smallest machine\[Hyphen]precision number which can be added to \
",
  StyleBox["1.0", "MR"],
  " to give a result that is distinguishable from ",
  StyleBox["1.0", "MR"],
  ". "
}], "Usage",
  CellTags->{"S0.1595", "$MachineEpsilon"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MachineEpsilon", "MR"],
  " is typically ",
  Cell[BoxData[
      \(TraditionalForm\`2\^\(-n + 1\)\)], "InlineFormula"],
  ", where ",
  Cell[BoxData[
      \(TraditionalForm\`n\)], "InlineFormula"],
  " is the number of binary bits used in the internal representation of \
machine\[Hyphen]precision floating\[Hyphen]point numbers. \n\
\[FilledSmallSquare] ",
  StyleBox["$MachineEpsilon", "MR"],
  " measures the granularity of machine\[Hyphen]precision numbers. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.1.6",
    ButtonData:>{"3.1.6", "6.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$MachinePrecision",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MinMachineNumber",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MaxMachineNumber",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
  ButtonBox["NumericalMath`ComputerArithmetic`",
    ButtonData:>{"NumericalMath`ComputerArithmetic`"},
    ButtonStyle->"AddOnsLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1595", "$MachineEpsilon"}],

Cell["$MachineID", "ObjectName",
  CellTags->{"S0.1596", "$MachineID"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MachineID", "MR"],
  " is a string which gives, if possible, a unique identification code for \
the computer on which ",
  StyleBox["Mathematica", "TI"],
  " is being run. "
}], "Usage",
  CellTags->{"S0.1596", "$MachineID"}],

Cell[TextData[{
  "\[FilledSmallSquare] On many computers, ",
  StyleBox["$MachineID", "MR"],
  " is the MathID string printed by the external program ",
  StyleBox["mathinfo", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.16"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$SystemID",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MachineName",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1596", "$MachineID"}],

Cell["$MachineName", "ObjectName",
  CellTags->{"S0.1597", "$MachineName"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MachineName", "MR"],
  " is a string which gives the assigned name of the computer on which ",
  StyleBox["Mathematica", "TI"],
  " is being run, if such a name is defined. "
}], "Usage",
  CellTags->{"S0.1597", "$MachineName"}],

Cell[TextData[{
  "\[FilledSmallSquare] For many classes of computers, ",
  StyleBox["$MachineName", "MR"],
  " is the network host name. \n\[FilledSmallSquare] ",
  StyleBox["$MachineName", "MR"],
  " is ",
  StyleBox["\"\"", "MR"],
  " if no name is defined. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.16"},
    ButtonStyle->"MainBookLink"],
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  ", ",
  ButtonBox["$System",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MachineID",
    ButtonStyle->"RefGuideLink"],
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Cell["$MachinePrecision", "ObjectName",
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Cell[TextData[{
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  ButtonBox["$MinMachineNumber",
    ButtonStyle->"RefGuideLink"],
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  ButtonBox["$MaxMachineNumber",
    ButtonStyle->"RefGuideLink"],
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    ButtonStyle->"AddOnsLink"],
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}], "Notes",
  CellTags->{"S0.1598", "$MachinePrecision"}],

Cell["$MachineType", "ObjectName",
  CellTags->{"S0.1599", "$MachineType"}],

Cell[TextData[{
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  CellTags->{"S0.1599", "$MachineType"}],

Cell[TextData[{
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  StyleBox["\"Sun\"", "MR"],
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  ", ",
  ButtonBox["$OperatingSystem",
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  CellTags->{"S0.1599", "$MachineType"}],

Cell["$MaxExtraPrecision", "ObjectName",
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Cell[TextData[{
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functions such as ",
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}], "Usage",
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Cell[TextData[{
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  Cell[BoxData[
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  StyleBox["=", "MR"],
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  Cell[BoxData[
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
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}], "Notes",
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Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
More than 50 extra digits are needed in intermediate calculations \
to get 20 digits in this result. \
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[4]:= ",
  CellTags->"$MaxExtraPrecision"],

Cell[BoxData[
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  CellLabel->"Out[4]= ",
  CellTags->"$MaxExtraPrecision"]
}, Open  ]],

Cell[TextData[{
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}], "ExampleText",
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"$MaxExtraPrecision"],

Cell[BoxData[
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  CellTags->"$MaxExtraPrecision"]
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Cell["\<\
Inequalities such as this one are evaluated using numerical \
methods. \
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"$MaxExtraPrecision"],

Cell[BoxData[
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  CellLabel->"Out[6]= ",
  CellTags->"$MaxExtraPrecision"]
}, Open  ]],

Cell["\<\
Some inequalities cannot be resolved numerically no matter how much \
extra precision is used. \
\>", "ExampleText",
  CellTags->"$MaxExtraPrecision"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[7]:= ",
  CellTags->"$MaxExtraPrecision"],

Cell[BoxData[
    \(1 + \@2 \[LessEqual] 1\/\(\(-1\) + \@2\)\)], "Output",
  CellLabel->"Out[7]= ",
  CellTags->"$MaxExtraPrecision"]
}, Open  ]],

Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"$MaxExtraPrecision"],

Cell[BoxData[
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  CellLabel->"Out[8]= ",
  CellTags->"$MaxExtraPrecision"]
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Cell["Clear the definition. ", "ExampleText",
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Cell[BoxData[
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  CellLabel->"In[9]:= ",
  CellTags->"$MaxExtraPrecision"],

Cell["$MaxMachineNumber", "ObjectName",
  CellTags->{"S0.1600", "$MaxMachineNumber"}],

Cell[TextData[{
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}], "Usage",
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Cell[TextData[{
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  " is the maximum exponent that can be used in the internal representation \
of machine\[Hyphen]precision numbers. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
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    ButtonStyle->"MainBookLink"],
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  ButtonBox["$MinMachineNumber",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MachineEpsilon",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MachinePrecision",
    ButtonStyle->"RefGuideLink"],
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  ButtonBox["NumericalMath`ComputerArithmetic`",
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    ButtonStyle->"AddOnsLink"],
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}], "Notes",
  CellTags->{"S0.1600", "$MaxMachineNumber"}],

Cell["$MaxNumber", "ObjectName",
  CellTags->{"S0.1600", "$MaxNumber"}],

Cell[TextData[{
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  StyleBox["$MaxNumber", "MR"],
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that can be represented on a particular computer system. "
}], "Usage",
  CellTags->{"S0.1600", "$MaxNumber"}],

Cell[TextData[{
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  Cell[BoxData[
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  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.1.6",
    ButtonData:>{"3.1.6", "6.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$MaxPrecision",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MaxMachineNumber",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1600", "$MaxNumber"}],

Cell["$MaxPrecision", "ObjectName",
  CellTags->{"S0.1600", "$MaxPrecision"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MaxPrecision", "MR"],
  " gives the maximum number of digits of precision to be allowed in \
arbitrary\[Hyphen]precision numbers. "
}], "Usage",
  CellTags->{"S0.1600", "$MaxPrecision"}],

Cell[TextData[{
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  " uses the maximum value possible on a particular computer system, given \
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  " ",
  StyleBox["$MaxNumber]", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$MaxPrecision", "MR"],
  " is measured in decimal digits, and need not be an integer. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.1.5",
    ButtonData:>{"3.1.5", "5.36"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$MinPrecision",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MaxExtraPrecision",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1600", "$MaxPrecision"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
  CellTags->"$MaxPrecision"],

Cell[TextData[{
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  StyleBox["$MaxPrecision", "MR"],
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}], "ExampleText",
  CellTags->"$MaxPrecision"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[1]:= ",
  CellTags->"$MaxPrecision"],

Cell[BoxData[
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  CellLabel->"Out[1]= ",
  CellTags->"$MaxPrecision"]
}, Open  ]],

Cell[TextData[{
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  StyleBox["$MaxPrecision", "MR"],
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}], "ExampleText",
  CellTags->"$MaxPrecision"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[2]:= ",
  CellTags->"$MaxPrecision"],

Cell[BoxData[
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current $MaxPrecision of \!\(50000.`\) instead. $MaxPrecision = Infinity \
specifies that any precision should be allowed."\)], "Message",
  CellTags->"$MaxPrecision"],

Cell[BoxData[
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  CellLabel->"Out[2]= ",
  CellTags->"$MaxPrecision"]
}, Open  ]],

Cell[TextData[{
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}], "ExampleText",
  CellTags->"$MaxPrecision"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[3]:= ",
  CellTags->"$MaxPrecision"],

Cell[BoxData[
    \(1000000\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"$MaxPrecision"]
}, Open  ]],

Cell[TextData[{
  StyleBox["$MaxPrecision", "MR"],
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high precision. "
}], "ExampleText",
  CellTags->"$MaxPrecision"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"$MaxPrecision"],

Cell[BoxData[
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  CellLabel->"Out[4]= ",
  CellTags->"$MaxPrecision"]
}, Open  ]],

Cell["$MessageList", "ObjectName",
  CellTags->{"S0.1601", "$MessageList"}],

Cell[TextData[{
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  StyleBox["$MessageList", "MR"],
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during the evaluation of the current input line. "
}], "Usage",
  CellTags->{"S0.1601", "$MessageList"}],

Cell[TextData[{
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",
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  StyleBox["{}", "MR"],
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\[FilledSmallSquare] You can reset ",
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  " during a computation. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.8.21",
    ButtonData:>{"2.8.21", "21.19"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["MessageList",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Check",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1601", "$MessageList"}],

Cell["$MessagePrePrint", "ObjectName",
  CellTags->{"S0.1602", "$MessagePrePrint"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MessagePrePrint", "MR"],
  " is a global variable whose value, if set, is applied to expressions \
before they are included in the text of messages. "
}], "Usage",
  CellTags->{"S0.1602", "$MessagePrePrint"}],

Cell[TextData[{
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  StyleBox["$MessagePrePrint", "MR"],
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  StyleBox["Short", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$MessagePrePrint", "MR"],
  " is applied after each expression is wrapped with ",
  StyleBox["HoldForm", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.8.21",
    ButtonData:>{"2.8.21", "21.15"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.13.1",
    ButtonData:>{"2.13.1", "1.33"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$PrePrint",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1602", "$MessagePrePrint"}],

Cell["$Messages", "ObjectName",
  CellTags->{"S0.1603", "$Messages"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$Messages", "MR"],
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}], "Usage",
  CellTags->{"S0.1603", "$Messages"}],

Cell[TextData[{
  "\[FilledSmallSquare] Output from ",
  StyleBox["Message", "MR"],
  " is always given on the ",
  StyleBox["$Messages", "MR"],
  " channel. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
    ButtonData:>{"2.13.1", "1.29"},
    ButtonStyle->"MainBookLink"],
  "."
}], "Notes",
  CellTags->{"S0.1603", "$Messages"}],

Cell["$MinMachineNumber", "ObjectName",
  CellTags->{"S0.1604", "$MinMachineNumber"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MinMachineNumber", "MR"],
  " is the smallest positive machine\[Hyphen]precision number that can be \
used on a particular computer system. "
}], "Usage",
  CellTags->{"S0.1604", "$MinMachineNumber"}],

Cell[TextData[{
  "\[FilledSmallSquare] See notes for ",
  ButtonBox["$MaxMachineNumber",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.1.6",
    ButtonData:>{"3.1.6", "6.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$MinNumber",
    ButtonStyle->"RefGuideLink"],
  ". \n\[FilledSmallSquare] Related package: ",
  ButtonBox["NumericalMath`ComputerArithmetic`",
    ButtonData:>{"NumericalMath`ComputerArithmetic`"},
    ButtonStyle->"AddOnsLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1604", "$MinMachineNumber"}],

Cell["$MinNumber", "ObjectName",
  CellTags->{"S0.1604", "$MinNumber"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MinNumber", "MR"],
  " gives the magnitude of the minimum positive arbitrary\[Hyphen]precision \
number that can be represented on a particular computer system. "
}], "Usage",
  CellTags->{"S0.1604", "$MinNumber"}],

Cell[TextData[{
  "\[FilledSmallSquare] A typical value for ",
  StyleBox["$MinNumber", "MR"],
  " is around ",
  Cell[BoxData[
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  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.1.6",
    ButtonData:>{"3.1.6", "6.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$MinPrecision",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MinMachineNumber",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1604", "$MinNumber"}],

Cell["$MinPrecision", "ObjectName",
  CellTags->{"S0.1604", "$MinPrecision"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$MinPrecision", "MR"],
  " gives the minimum number of digits of precision to be allowed in \
arbitrary\[Hyphen]precision numbers. "
}], "Usage",
  CellTags->{"S0.1604", "$MinPrecision"}],

Cell[TextData[{
  "\[FilledSmallSquare] The default value of ",
  StyleBox["$MinPrecision", "MR"],
  " is 0. \n\[FilledSmallSquare] Positive values of ",
  StyleBox["$MinPrecision", "MR"],
  " make ",
  StyleBox["Mathematica", "TI"],
  " pad arbitrary\[Hyphen]precision numbers with zero digits to achieve the \
specified nominal precision. The zero digits are taken to be in base 2, and \
may not correspond to zeros in base 10. \n\[FilledSmallSquare] ",
  StyleBox["$MaxPrecision", "MR"],
  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["$MinPrecision", "MR"],
  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["n", "TI"],
  " makes ",
  StyleBox["Mathematica", "TI"],
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  StyleBox["$MinPrecision", "MR"],
  " is measured in decimal digits, and need not be an integer. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.1.5",
    ButtonData:>{"3.1.5", "5.36"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["SetPrecision",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MinNumber",
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
To achieve certain special effects, it is sometimes useful to set a \
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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value of ",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["$ModuleNumber", "ObjectName",
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Cell[TextData[{
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Cell["$NewMessage", "ObjectName",
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Cell["$NewSymbol", "ObjectName",
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Cell[TextData[{
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Cell["$Notebooks", "ObjectName",
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Cell["$NumberMarks", "ObjectName",
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\[FilledSmallSquare] ",
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  StyleBox["=", "MR"],
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  StyleBox["Automatic", "MR"],
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\[Hyphen]precision numbers. \n\[FilledSmallSquare] ",
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  StyleBox["=", "MR"],
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\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
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  ButtonBox["SetPrecision",
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}], "Notes",
  CellTags->{"S0.1608", "$NumberMarks"}],

Cell["$OperatingSystem", "ObjectName",
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Cell[TextData[{
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  StyleBox["Mathematica", "TI"],
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}], "Usage",
  CellTags->{"S0.1609", "$OperatingSystem"}],

Cell[TextData[{
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  StyleBox["\"MSDOS\"", "MR"],
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  StyleBox["\"MacOS\"", "MR"],
  ", ",
  StyleBox["\"Unix\"", "MR"],
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  StyleBox["\"VMS\"", "MR"],
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  StyleBox["Mathematica", "TI"],
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
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  ".\n\[FilledSmallSquare] See also: ",
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  ButtonBox["$System",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$ProcessorType",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1609", "$OperatingSystem"}],

Cell["$Output", "ObjectName",
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Cell[TextData[{
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}], "Usage",
  CellTags->{"S0.1610", "$Output"}],

Cell[TextData[{
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  StyleBox["$Output", "MR"],
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
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  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Streams",
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}], "Notes",
  CellTags->{"S0.1610", "$Output"}],

Cell["$Packages", "ObjectName",
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Cell[TextData[{
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been loaded in your current ",
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}], "Usage",
  CellTags->{"S0.1611", "$Packages"}],

Cell[TextData[{
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  StyleBox["BeginPackage", "MR"],
  " is executed. \n\[FilledSmallSquare] ",
  StyleBox["$Packages", "MR"],
  " is used by ",
  StyleBox["Needs", "MR"],
  " to determine whether a particular package needs to be loaded explicitly. \
\n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.6.9",
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  ButtonBox["$ContextPath",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["DeclarePackage",
    ButtonStyle->"RefGuideLink"],
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  ButtonBox["$ExportFormats",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1611", "$Packages"}],

Cell["$ParentLink", "ObjectName",
  CellTags->{"S0.1611", "$ParentLink"}],

Cell[TextData[{
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  StyleBox["MathLink", "TI"],
  " ",
  StyleBox["LinkObject", "MR"],
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}], "Usage",
  CellTags->{"S0.1611", "$ParentLink"}],

Cell[TextData[{
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  StyleBox["Mathematica", "TI"],
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  StyleBox["$ParentLink", "MR"],
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  StyleBox["MathLink", "TI"],
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You can reset ",
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\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.12.8",
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    ButtonStyle->"MainBookLink"],
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  ", ",
  ButtonBox["Links",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Input",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1611", "$ParentLink"}],

Cell["$ParentProcessID", "ObjectName",
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Cell[TextData[{
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  StyleBox["Mathematica", "TI"],
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}], "Usage",
  CellTags->{"S0.1611", "$ParentProcessID"}],

Cell[TextData[{
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",
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  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
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    ButtonStyle->"MainBookLink"],
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  ", ",
  ButtonBox["$ProcessID",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$CommandLine",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$UserName",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Environment",
    ButtonStyle->"RefGuideLink"],
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}], "Notes",
  CellTags->{"S0.1611", "$ParentProcessID"}],

Cell["$Path", "ObjectName",
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Cell[TextData[{
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}], "Usage",
  CellTags->{"S0.1612", "$Path"}],

Cell[TextData[{
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strings. The full file names tested are of the form ",
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  StyleBox["directory", "TI"],
  StyleBox[",", "MR"],
  StyleBox["name", "TI"],
  StyleBox["]", "MR"],
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characters can be used in directory names: "
}], "Notes",
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Cell[BoxData[
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          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"..\\\\\\\"\\\", \\\"MR\\\"]\\) \
\"\>", "\<\"the directory one level up in the hierarchy \"\>"},
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"~\\\\\\\"\\\", \\\"MR\\\"]\\) \
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Cell[TextData[{
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  StyleBox["The Mathematica Book: ", "TI"],
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  ButtonBox["Directory",
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  ", ",
  ButtonBox["SetDirectory",
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  ButtonBox["Get",
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  ", ",
  ButtonBox["DirectoryName",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Input",
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}], "Notes",
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Cell["$Post", "ObjectName",
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Cell[TextData[{
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  CellTags->{"S0.1613", "$Post"}],

Cell[TextData[{
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  ButtonBox["Section 2.13.1",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$Pre",
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  ", ",
  ButtonBox["$PrePrint",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1613", "$Post"}],

Cell["$Pre", "ObjectName",
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Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$Pre", "MR"],
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expression. "
}], "Usage",
  CellTags->{"S0.1614", "$Pre"}],

Cell[TextData[{
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  StyleBox["$Pre", "MR"],
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  StyleBox["$Post", "MR"],
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  StyleBox["$Pre", "MR"],
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  StyleBox["$PreRead", "MR"],
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\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
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    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$Post",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1614", "$Pre"}],

Cell["$PrePrint", "ObjectName",
  CellTags->{"S0.1615", "$PrePrint"}],

Cell[TextData[{
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  StyleBox["$PrePrint", "MR"],
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before it is printed. "
}], "Usage",
  CellTags->{"S0.1615", "$PrePrint"}],

Cell[TextData[{
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  StyleBox["n", "TI"],
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\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
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    ButtonStyle->"MainBookLink"],
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  ButtonBox["$Post",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MessagePrePrint",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1615", "$PrePrint"}]
}, Closed]],

Cell[CellGroupData[{

Cell["Further Examples", "ExampleSection",
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Cell[TextData[{
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  StyleBox["Pi", "MR"],
  " and ",
  StyleBox["E", "MR"],
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}], "ExampleText",
  CellTags->"$PrePrint"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellTags->"$PrePrint"],

Cell[BoxData[
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  CellLabel->"Out[1]= ",
  CellTags->"$PrePrint"]
}, Open  ]],

Cell[TextData[{
  "This setting of ",
  StyleBox["$PrePrint", "MR"],
  " make numbers that appear in output approximate. "
}], "ExampleText",
  CellTags->"$PrePrint"],

Cell[CellGroupData[{

Cell[BoxData[
    \($PrePrint = N\)], "Input",
  CellLabel->"In[2]:= ",
  CellTags->"$PrePrint"],

Cell[BoxData[
    \(N\)], "Output",
  CellLabel->"Out[2]= ",
  CellTags->"$PrePrint"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \({\[Pi], \[ExponentialE]}\)], "Input",
  CellLabel->"In[3]:= ",
  CellTags->"$PrePrint"],

Cell[BoxData[
    \({3.141592653589793`, 2.718281828459045`}\)], "Output",
  CellLabel->"Out[3]= ",
  CellTags->"$PrePrint"]
}, Open  ]],

Cell["This restores the original behavior. ", "ExampleText",
  CellTags->"$PrePrint"],

Cell[BoxData[
    \($PrePrint =  . \)], "Input",
  CellLabel->"In[4]:= ",
  CellTags->"$PrePrint"],

Cell[CellGroupData[{

Cell[BoxData[
    \({\[Pi], \[ExponentialE]}\)], "Input",
  CellLabel->"In[5]:= ",
  CellTags->"$PrePrint"],

Cell[BoxData[
    \({\[Pi], \[ExponentialE]}\)], "Output",
  CellLabel->"Out[5]= ",
  CellTags->"$PrePrint"]
}, Open  ]],

Cell["$PreRead", "ObjectName",
  CellTags->{"S0.1616", "$PreRead"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$PreRead", "MR"],
  " is a global variable whose value, if set, is applied to the text or box \
form of every input expression before it is fed to ",
  StyleBox["Mathematica", "TI"],
  ". "
}], "Usage",
  CellTags->{"S0.1616", "$PreRead"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$PreRead", "MR"],
  " is always applied to each complete input string that will be fed to ",
  StyleBox["Mathematica", "TI"],
  ". \n\[FilledSmallSquare] In multiline input with a text\[Hyphen]based \
interface, ",
  StyleBox["$PreRead", "MR"],
  " is typically applied to the input so far whenever each line is \
terminated. \n\[FilledSmallSquare] ",
  StyleBox["$PreRead", "MR"],
  " is applied to all strings returned by a ",
  StyleBox["$SyntaxHandler", "MR"],
  " function. \n\[FilledSmallSquare] ",
  StyleBox["$PreRead", "MR"],
  " is applied before ",
  StyleBox["InString[", "MR"],
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
  " is assigned. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
    ButtonData:>{"2.13.1", "1.12"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$Pre",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["StringReplace",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ToExpression",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1616", "$PreRead"}],

Cell["$ProcessID", "ObjectName",
  CellTags->{"S0.1616", "$ProcessID"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$ProcessID", "MR"],
  " gives the ID assigned to the ",
  StyleBox["Mathematica", "TI"],
  " kernel process by the operating system under which it is run. "
}], "Usage",
  CellTags->{"S0.1616", "$ProcessID"}],

Cell[TextData[{
  "\[FilledSmallSquare] On operating systems where no process ID is assigned, \
",
  StyleBox["$ProcessID", "MR"],
  " is ",
  StyleBox["None", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.7"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$ParentProcessID",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$UserName",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Environment",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$SessionID",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1616", "$ProcessID"}],

Cell["$ProcessorType", "ObjectName",
  CellTags->{"S0.1616", "$ProcessorType"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$ProcessorType", "MR"],
  " is a string giving the architecture of processor on which ",
  StyleBox["Mathematica", "TI"],
  " is being run. "
}], "Usage",
  CellTags->{"S0.1616", "$ProcessorType"}],

Cell[TextData[{
  "\[FilledSmallSquare] Typical values are ",
  StyleBox["\"x86\"", "MR"],
  ", ",
  StyleBox["\"680x0\"", "MR"],
  ", ",
  StyleBox["\"SPARC\"", "MR"],
  ", ",
  StyleBox["\"MIPS\"", "MR"],
  ", ",
  StyleBox["\"PA-RISC\"", "MR"],
  " and ",
  StyleBox["\"AXP\"", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$ProcessorType", "MR"],
  " specifies the basic instruction set used by the CPU of your computer. \n\
\[FilledSmallSquare] Computers with the same ",
  StyleBox["$ProcessorType", "MR"],
  " may not be binary compatible. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.13"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$MachineType",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$OperatingSystem",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$System",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1616", "$ProcessorType"}],

Cell["$RandomState", "ObjectName",
  CellTags->{"S0.1616", "$RandomState"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$RandomState", "MR"],
  " gives a representation of the internal state of the pseudorandom \
generator used by ",
  StyleBox["Random", "MR"],
  ". "
}], "Usage",
  CellTags->{"S0.1616", "$RandomState"}],

Cell[TextData[{
  "\[FilledSmallSquare] The value of ",
  StyleBox["$RandomState", "MR"],
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  StyleBox["Random", "MR"],
  " is called. \n\[FilledSmallSquare] You can use ",
  StyleBox["s", "TI"],
  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["$RandomState", "MR"],
  " to explicitly save the value of ",
  StyleBox["$RandomState", "MR"],
  ", and ",
  StyleBox["$RandomState", "MR"],
  " ",
  StyleBox["=", "MR"],
  " ",
  StyleBox["s", "TI"],
  " to restore. \n\[FilledSmallSquare] The value of ",
  StyleBox["$RandomState", "MR"],
  " is always a long integer chosen from a certain large set of \
possibilities. You can assign ",
  StyleBox["$RandomState", "MR"],
  " only to values in this set. \n\[FilledSmallSquare] You can use ",
  StyleBox["Block[", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`{\)], "InlineFormula"],
  StyleBox["$RandomState", "MR"],
  Cell[BoxData[
      \(TraditionalForm\`}\)], "InlineFormula"],
  StyleBox[",", "MR"],
  " ",
  StyleBox["expr", "TI"],
  StyleBox["]", "MR"],
  " to localize the value of ",
  StyleBox["$RandomState", "MR"],
  " during the evaluation of ",
  StyleBox["expr", "TI"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 3.2.3",
    ButtonData:>{"3.2.3", "3.2"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["SeedRandom",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1616", "$RandomState"}],

Cell["$RecursionLimit", "ObjectName",
  CellTags->{"S0.1617", "$RecursionLimit"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$RecursionLimit", "MR"],
  " gives the current limit on the number of levels of recursion that ",
  StyleBox["Mathematica", "TI"],
  " can use. "
}], "Usage",
  CellTags->{"S0.1617", "$RecursionLimit"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$RecursionLimit=", "MR"],
  StyleBox["n", "TI"],
  " sets the limit on the number of recursion levels that ",
  StyleBox["Mathematica", "TI"],
  " can use to be ",
  StyleBox["n", "TI"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$RecursionLimit=Infinity", "MR"],
  " removes any limit on the number of recursion levels. \n\
\[FilledSmallSquare] ",
  StyleBox["$RecursionLimit", "MR"],
  " gives the maximum length of the stack returned by ",
  StyleBox["Stack[", "MR"],
  " ",
  StyleBox["]", "MR"],
  ". \n\[FilledSmallSquare] Each time the evaluation of a function requires \
the nested evaluation of the same or another function, one recursion level is \
used up. \n\[FilledSmallSquare] On most computers, each level of recursion \
uses a certain amount of stack space. ",
  StyleBox["$RecursionLimit", "MR"],
  " allows you to control the amount of stack space that ",
  StyleBox["Mathematica", "TI"],
  " can use from within ",
  StyleBox["Mathematica", "TI"],
  ". On some computer systems, your whole ",
  StyleBox["Mathematica", "TI"],
  " session may crash if you allow it to use more stack space than the \
computer system allows. \n\[FilledSmallSquare] ",
  StyleBox["MemoryInUse", "MR"],
  " and related functions do not count stack space. \n\[FilledSmallSquare] \
See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.5.12",
    ButtonData:>{"2.5.12", "12.2"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.5.12",
    ButtonData:>{"2.5.12", "12.5"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$IterationLimit",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["MemoryConstrained",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1617", "$RecursionLimit"}],

Cell["$ReleaseNumber", "ObjectName",
  CellTags->{"S0.1618", "$ReleaseNumber"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$ReleaseNumber", "MR"],
  " is an integer which gives the current ",
  StyleBox["Mathematica", "TI"],
  " kernel release number, and increases in successive releases. "
}], "Usage",
  CellTags->{"S0.1618", "$ReleaseNumber"}],

Cell[TextData[{
  "\[FilledSmallSquare] Each released revision of the ",
  StyleBox["Mathematica", "TI"],
  " kernel for any particular computer system is assigned a new release \
number. \n\[FilledSmallSquare] The same source code may yield releases with \
different numbers on different computer systems. \n\[FilledSmallSquare] See \
",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.11"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$VersionNumber",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1618", "$ReleaseNumber"}],

Cell["$SessionID", "ObjectName",
  CellTags->{"S0.1619", "$SessionID"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$SessionID", "MR"],
  " is a number set up to be unique to a particular ",
  StyleBox["Mathematica", "TI"],
  " session. "
}], "Usage",
  CellTags->{"S0.1619", "$SessionID"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$SessionID", "MR"],
  " should be different for different ",
  StyleBox["Mathematica", "TI"],
  " sessions run either on the same computer or on different computers. \n\
\[FilledSmallSquare] The value of ",
  StyleBox["$SessionID", "MR"],
  " is based on ",
  StyleBox["$MachineID", "MR"],
  ", as well as ",
  StyleBox["AbsoluteTime[", "MR"],
  " ",
  StyleBox["]", "MR"],
  " and operating system parameters such as the ",
  StyleBox["Mathematica", "TI"],
  " process ID. \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.6.3",
    ButtonData:>{"2.6.3", "3.26"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.9"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$ModuleNumber",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$ProcessID",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1619", "$SessionID"}],

Cell["$SoundDisplayFunction", "ObjectName",
  CellTags->{"S0.1620", "$SoundDisplayFunction"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$SoundDisplayFunction", "MR"],
  " gives the default setting for the option ",
  StyleBox["DisplayFunction", "MR"],
  " in sound functions. "
}], "Usage",
  CellTags->{"S0.1620", "$SoundDisplayFunction"}],

Cell[TextData[{
  "\[FilledSmallSquare] The initial setting of ",
  StyleBox["$SoundDisplayFunction", "MR"],
  " is ",
  StyleBox["Display[$SoundDisplay,", "MR"],
  " ",
  StyleBox["#]&", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.9.18",
    ButtonData:>{"2.9.18", "18.10"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["Play",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["ListPlay",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["Show",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$DisplayFunction",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1620", "$SoundDisplayFunction"}],

Cell["$SyntaxHandler", "ObjectName",
  CellTags->{"S0.1621", "$SyntaxHandler"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$SyntaxHandler", "MR"],
  " is a global variable which, if set, is applied to any input string that \
is found to contain a syntax error. "
}], "Usage",
  CellTags->{"S0.1621", "$SyntaxHandler"}],

Cell[TextData[{
  "\[FilledSmallSquare] The arguments given to ",
  StyleBox["$SyntaxHandler", "MR"],
  " are the complete input string and an integer specifying the character \
position at which the syntax error was detected. \n\[FilledSmallSquare] The \
first character in the string is taken to have position ",
  StyleBox["1", "MR"],
  ". \n\[FilledSmallSquare] Any string returned by ",
  StyleBox["$SyntaxHandler", "MR"],
  " is used as a new version of the input string, and is fed to ",
  StyleBox["Mathematica", "TI"],
  ". If the string does not end with a newline, ",
  StyleBox["Mathematica", "TI"],
  " waits for input to complete the line. \n\[FilledSmallSquare] If ",
  StyleBox["$SyntaxHandler", "MR"],
  " returns ",
  StyleBox["$Failed", "MR"],
  ", input to ",
  StyleBox["Mathematica", "TI"],
  " is abandoned if possible. \n\[FilledSmallSquare] Input is not assigned to \
",
  StyleBox["InString[", "MR"],
  StyleBox["n", "TI"],
  StyleBox["]", "MR"],
  " until after ",
  StyleBox["$SyntaxHandler", "MR"],
  " is applied. \n\[FilledSmallSquare] ",
  StyleBox["$SyntaxHandler", "MR"],
  " is not called for input from files obtained using ",
  StyleBox["Get", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.1",
    ButtonData:>{"2.13.1", "1.12"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["SyntaxLength",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["SyntaxQ",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1621", "$SyntaxHandler"}],

Cell["$System", "ObjectName",
  CellTags->{"S0.1622", "$System"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$System", "MR"],
  " is a string describing the type of computer system on which ",
  StyleBox["Mathematica", "TI"],
  " is being run. "
}], "Usage",
  CellTags->{"S0.1622", "$System"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$System", "MR"],
  " typically consists of words separated by spaces. \n\[FilledSmallSquare] \
Typical values are ",
  StyleBox["\"Microsoft", "MR"],
  " ",
  StyleBox["Windows\"", "MR"],
  ", ",
  StyleBox["\"Macintosh\"", "MR"],
  ", ",
  StyleBox["\"SPARC\"", "MR"],
  " and ",
  StyleBox["\"Silicon", "MR"],
  " ",
  StyleBox["Graphics\"", "MR"],
  ". \n\[FilledSmallSquare] ",
  StyleBox["$SystemID", "MR"],
  " provides a more succinct version of the same information. \n\
\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.13"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$SystemID",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Version",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MachineType",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$ProcessorType",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$OperatingSystem",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1622", "$System"}],

Cell["$SystemCharacterEncoding", "ObjectName",
  CellTags->{"S0.1622", "$SystemCharacterEncoding"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$SystemCharacterEncoding", "MR"],
  " gives the default raw character encoding for the computer system on which \
",
  StyleBox["Mathematica", "TI"],
  " is being run. "
}], "Usage",
  CellTags->{"S0.1622", "$SystemCharacterEncoding"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$SystemCharacterEncoding", "MR"],
  " is used to determine the default value of ",
  StyleBox["$CharacterEncoding", "MR"],
  ". \n\[FilledSmallSquare] The notebook front end handles raw character \
encodings independent of the kernel. \n\[FilledSmallSquare] The possible \
settings for ",
  StyleBox["$SystemCharacterEncoding", "MR"],
  " are the same as for ",
  StyleBox["$CharacterEncoding", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.7.8",
    ButtonData:>{"2.7.8", "8.3"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["CharacterEncoding",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1622", "$SystemCharacterEncoding"}],

Cell["$SystemID", "ObjectName",
  CellTags->{"S0.1622", "$SystemID"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$SystemID", "MR"],
  " is a short string that identifies the type of computer system on which ",
  StyleBox["Mathematica", "TI"],
  " is being run. "
}], "Usage",
  CellTags->{"S0.1622", "$SystemID"}],

Cell[TextData[{
  "\[FilledSmallSquare] Computer systems with the same ",
  StyleBox["$SystemID", "MR"],
  " should be binary compatible, so that the same external programs and ",
  StyleBox[".mx", "MR"],
  " files can be used. \n\[FilledSmallSquare] Sometimes binary compatibility \
may only be complete when the same version of the operating system is used. \n\
\[FilledSmallSquare] ",
  StyleBox["$SystemID", "MR"],
  " is used in naming directories generated by ",
  StyleBox["DumpSave", "MR"],
  " and ",
  StyleBox["mcc", "MR"],
  ". \n\[FilledSmallSquare] Values for ",
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  " contain only alphanumeric characters and dashes. \n\[FilledSmallSquare] \
Typical values are ",
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  ", ",
  StyleBox["\"Mac\"", "MR"],
  ", ",
  StyleBox["\"SPARC\"", "MR"],
  " and ",
  StyleBox["\"SGI\"", "MR"],
  ". \n\[FilledSmallSquare] See ",
  StyleBox["The Mathematica Book: ", "TI"],
  ButtonBox["Section 2.11.1",
    ButtonData:>{"2.11.1", "1.38"},
    ButtonStyle->"MainBookLink"],
  ", ",
  ButtonBox["Section 2.12.5",
    ButtonData:>{"2.12.5", "5.3"},
    ButtonStyle->"MainBookLink"],
  " and\[NonBreakingSpace]",
  ButtonBox["Section 2.13.5",
    ButtonData:>{"2.13.5", "5.13"},
    ButtonStyle->"MainBookLink"],
  ".\n\[FilledSmallSquare] See also: ",
  ButtonBox["$System",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$Version",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$MachineType",
    ButtonStyle->"RefGuideLink"],
  ", ",
  ButtonBox["$OperatingSystem",
    ButtonStyle->"RefGuideLink"],
  ". "
}], "Notes",
  CellTags->{"S0.1622", "$SystemID"}],

Cell["$TextStyle", "ObjectName",
  CellTags->{"S0.1622", "$TextStyle"}],

Cell[TextData[{
  "\[FilledSmallSquare] ",
  StyleBox["$TextStyle", "MR"],
  " gives the default style to use for text in graphics. "
}], "Usage",
  CellTags->{"S0.1622", "$TextStyle"}],

Cell["\<\
\[FilledSmallSquare] The following forms of settings can be used: \
\
\>", "Notes",
  CellTags->{"S0.1622", "$TextStyle"}],

Cell[BoxData[
    FormBox[GridBox[{
          {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\\\\\\\"\\\", \
\\\"MR\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"style\\\\\\\"\\\", \\\"TI\\\"]\\)\
\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\) \"\
\>", "\<\"a specific cell style \"\>"},
          {
            RowBox[{
              StyleBox["{",
                "InlineFormula"], 
              StyleBox[
                
                SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"opt\\\\\\\"\\\
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                "InlineFormula"], "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"->\\\\\\\
\"\\\", \\\"MR\\\"]\\)\"\>", 
              StyleBox[
                
                SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"val\\\\\\\"\\\
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                "InlineFormula"], \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \[Ellipsis] \
\"\>", 
              StyleBox["}",
                
                "InlineFormula"], "\<\" \"\>"}], "\<\"a list of option \
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          {
            RowBox[{
              StyleBox["{",
                
                "InlineFormula"], "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\
\\\\\\\"\\\\\\\"\\\", \\\"MR\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"style\\\\\\\
\"\\\", \\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"\\\\\\\\\\\\\\\",\\\\\\\"\
\\\", \\\"MR\\\"]\\) \"\>", 
              StyleBox[
                
                SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"opt\\\\\\\"\\\
\", \\\"TI\\\"]\\)\"\>", "1"],
                
                "InlineFormula"], "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"->\\\\\\\
\"\\\", \\\"MR\\\"]\\)\"\>", 
              StyleBox[
                
                SubscriptBox["\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"val\\\\\\\"\\\
\", \\\"TI\\\"]\\)\"\>", "1"],
                
                "InlineFormula"], \
"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \[Ellipsis] \
\"\>", 
              StyleBox["}",
                
                "InlineFormula"], "\<\" \"\>"}], "\<\"a style modified by \
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          },
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*)

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*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)

