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Notebook[{

Cell[CellGroupData[{
Cell["The package OscillatorPlot - 2", "Title"],

Cell["\<\
Bernd Thaller
Institute for Mathematics
University of Graz\
\>", "Subsubsection"],

Cell[CellGroupData[{

Cell["Defining a package context", "Section"],

Cell[TextData[{
  "As a next step, we put the commands exported by the package in a \"package \
context\". We name this package context \"Graphics`OscillatorPlot`\". Using \
the BeginPackage - EndPackage commands ensures that the package context is \
put on the context search path after loading the package. If you place the \
file OscillatorPlot2.m in the Graphics-subdirectory of AddOns/Applications, \
then it is automatically on ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s search path. You can then load it with the command \
<<Graphics`OscillatorPlot2`."
}], "Text"],

Cell[TextData[{
  "BeginPackage[\"Graphics`OscillatorPlot`\"]\n\n\
OscillatorPlot::usage=\"OscillatorPlot[f[x],{x,a,b},n1,n2] plots the energy \
representation of a given function in the basis of harmonic oscillator \
eigenfunctions. Numerical region for the determination of the expansion\n\
coefficients is the interval (a,b). The graph shows the expansion \
coefficients c_n\nfor n between n1 and n2. Default values are: a=-Infinity, \
b=Infinity, n1=0, n2=10.\"\n\nBegin[\"`Private`\"]\n",
  StyleBox["\n",
    FontVariations->{"CompatibilityType"->0}],
  "OscillatorFunction[n_,x_] :=\n   \
HermiteH[n,x]*Exp[-x^2/2]/Sqrt[2^n*n!]/Pi^(1/4)\n",
  StyleBox["\n",
    FontVariations->{"CompatibilityType"->0}],
  "OscillatorPlot[f_,{x_Symbol,x1_:-Infinity,x2_:Infinity},\n               \
n1_:0,n2_:10]:=\n   Module[{coeff,coloredLine,lineTable,up,upper,lower,func},\
\n      func=Function[{x},f];\n      Do[coeff[n]=\n         \
NIntegrate[OscillatorFunction[n,x]*func[x],{x,x1,x2}],{n,n1,n2}];\n      \
coloredLine[n_]:={Thickness[0.01],Hue[Arg[coeff[n]]/(2*Pi)],\n                \
     Line[{{n+1/2,Abs[coeff[n]]},{n+1/2,0.}}]};\n      \
lineTable=Graphics[Table[coloredLine[n],{n,n1,n2}]];\n      \
up=Max[Table[Abs[coeff[n]],{n,n1,n2}]];\n      upper=up+up/10; \
lower=-(up/10);\n      Show[lineTable, Frame\[Rule]True, \
PlotRange\[Rule]{{n1,n2+1}, {lower,upper}}]\n   ]\n   \nEnd[]\n\n\
EndPackage[]"
}], "Program"],

Cell[TextData[{
  "\n",
  StyleBox["Start a new kernel session here!",
    FontWeight->"Bold"],
  "\n\nNow the package can be loaded with"
}], "Text"],

Cell[BoxData[
    \(<< Graphics`OscillatorPlot2`\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \($ContextPath\)], "Input"],

Cell[BoxData[
    \({"Graphics`OscillatorPlot`", "Global`", "System`"}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?OscillatorPlot\)\)], "Input"],

Cell[BoxData[
    \("OscillatorPlot[f[x],{x,a,b},n1,n2] plots the energy representation of \
a given function in the basis of harmonic oscillator eigenfunctions. \
Numerical region for the determination of the expansion coefficients is the \
interval (a,b). The graph shows the expansion coefficients c_n for n between \
n1 and n2. Default values are: a=-Infinity, b=Infinity, n1=0, n2=10."\)], \
"Print"]
}, Open  ]],

Cell["\<\
Presently, the symbol OscillatorFunction is in the private context \
of the package. This context is not searched during the main session. \
Therefore this symbol would not conflict with any user-defined \
OscillatorFunction. The symbol OscillatorPlot is in the context \
Graphics`OscillatorPlot` which is on the context path. Hence it was necessary \
to start a new session before loading the package. Otherwise the earlier \
defined symbol OscillatorPlot (in the global context) would have shadowed the \
symbol in the package.\
\>", "Text"],

Cell[TextData[{
  "Note: You do not need to store the package files in the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " folder. During development, it is better to place the files somewhere. \
Instead of worrying whether \"somewhere\" is on the search path, we keep the \
file in an open notebook window and load the package just by activating the \
window and evaluating the notebook."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Default values", "Section"],

Cell["\<\
The package OscillatorPlot2 already defines defaults in order to \
facilitate the input. Since the defaults are simple numbers, we can put it \
just into the input-pattern\
\>", "Text"],

Cell[BoxData[
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Cell[TextData[{
  "This pattern matches any expression, later referred to as \"n1\". If no \
input is given for n1, then the value 0 is automatically assumed. ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " gives you several possibilities to define default values for arguments of \
functions. Let us quote here the text from the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " book:"
}], "Text"],

Cell[TextData[{
  "When the pattern objects ",
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  StyleBox["v", "TI"],
  " and ",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
In the package OscillatorPlot, we use default values for the \
numerical region and for the range of indices. Thus you could simply type \
OscillatorPlot[Exp[-x^2],{x}] instead of \
OscillatorPlot[[Exp[-x^2],{x,-Infinity, Infinity}, 0,10]. If you specify \
other values, you have to type the input using the longer form.\
\>", "Text"],

Cell[CellGroupData[{

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