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MathGroup Archive 1997

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Re: Integrating by parts....II

  • To: mathgroup at smc.vnet.net
  • Subject: [mg8786] Re: Integrating by parts....II
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 25 Sep 1997 12:26:20 -0400
  • Organization: University of Western Australia
  • Sender: owner-wri-mathgroup at wolfram.com

Adrean Webb wrote:

> Can Mathematica still solve the function below?
> 
>         f[n] = Integrate[Cos[x]*E[-x^2]*HermiteH[n,x],{x,-Inf,Inf}]

Yes.  You can compute the set of integrals,

	Integrate[Cos[x]*E[-x^2]*HermiteH[n,x],{x,-Inf,Inf}]

for arbitrary n using the Hermite generating function.  See the Notebook
below for details. (Select from Notebook[  down to ] inclusive and paste
into a new Mathematica Notebook).

Cheers,
	Paul 

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia           
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au 
AUSTRALIA                             http://www.pd.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________



Notebook[{
Cell[TextData[{
  "Using the Hermite generating function, ",
  Cell[BoxData[
      \(TraditionalForm
      \`\(\[ScriptCapitalH](x, z) == 
        \[Sum]\+\(n = 0\)\%\[Infinity]\(\(\( H\_n\)(x)\)\
z\^n\)\/\(n!\), 
      \)\)]]
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(\[ScriptCapitalH](x_, z_) = \[ExponentialE]\^\(2\ x\ z - z\^2\); 
    \)\)], "Input",
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`\(\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity] 
                \[ExponentialE]\^\(-x\^2\)\ \(cos(x)\)\ 
              \(\[ScriptCapitalH](x, z)\) \[DifferentialD]x // Factor\)
// 
        ComplexExpand\) // TrigReduce\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(\ at \[Pi]\ \(cos(z)\)\)\/\ at \[ExponentialE]\%4\)], 
  "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{"%", 
        FormBox[\(+\(O(z)\)\^5\),
          "TraditionalForm"]}], TraditionalForm]], "Input"],

Cell[BoxData[
    FormBox[
      InterpretationBox[
        RowBox[{
        \(\ at \[Pi]\/\ at \[ExponentialE]\%4\), "-", 
          \(\(\ at \[Pi]\ z\^2\)\/\(2\ \ at \[ExponentialE]\%4\)\), "+", 
          \(\(\ at \[Pi]\ z\^4\)\/\(24\ \ at \[ExponentialE]\%4\)\), "+", 
          InterpretationBox[\(O(z\^5)\),
            SeriesData[ z, 0, {}, 0, 5, 1]]}],
        SeriesData[ z, 0, {
          Times[ 
            Power[ E, 
              Rational[ -1, 4]], 
            Power[ Pi, 
              Rational[ 1, 2]]], 0, 
          Times[ 
            Rational[ -1, 2], 
            Power[ E, 
              Rational[ -1, 4]], 
            Power[ Pi, 
              Rational[ 1, 2]]], 0, 
          Times[ 
            Rational[ 1, 24], 
            Power[ E, 
              Rational[ -1, 4]], 
            Power[ Pi, 
              Rational[ 1, 2]]]}, 0, 5, 1]], TraditionalForm]],
"Output"]
}, Open  ]]
}
]


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