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Re: Integrating special functions

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  • Subject: [mg131599] Re: Integrating special functions
  • From: Alexei Boulbitch <Alexei.Boulbitch at>
  • Date: Tue, 10 Sep 2013 03:34:48 -0400 (EDT)
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Dear All,

I would like to integrate the following function with Legendre polynomial and Gamma function. I am open to any suggestions.

 ass = {s > 0, \[Alpha] > 0};

 \[Phi][s_, x_, \[Alpha]_] := (-1)^s Sqrt[(s \[Alpha])/
  Gamma[1 + 2*s]] LegendreP[s, s, Tanh[\[Alpha] x]]

\[Phi]1[s_, x_, \[Alpha]_] := D[[\[Phi][s, x, \[Alpha]], x]]

a3 = Table[
  Integrate[-\[ImaginaryI] x \[Phi][s, x, a] \[Phi]1[s, x,
     a], {x, -\[Infinity], \[Infinity]}, Assumptions -> \[Alpha] > 0,
   s > 0]
would it be possible to get a closed form of the integration a3?

Hi, Herman,

Since f(x)*f'(x)=g'(x) where g(x)=0.5*[f(x)]^2, your integral is easily transformed by parts into form:


You may add any limits to the above integral. The last term above is:

int=Integrate[LegendreP[s, s,Tanh[x]]^2, {x, -\[Infinity], \[Infinity]}]

It is approximately int=Exp[-1.025+1.303*s^3/2] as I have shown here:

Have fun, Alexei

Alexei BOULBITCH, Dr., habil.
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