Re: Integrating special functions

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• Subject: [mg131599] Re: Integrating special functions
• From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
• 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:

Integrate[x*g'[x],x]===x*g(x)-Integrate[g[x],x]

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:

http://mathematica.stackexchange.com/questions/31534/integrating-special-functions/31776?noredirect=1#comment99086_31776

Have fun, Alexei

Alexei BOULBITCH, Dr., habil.
IEE S.A.
ZAE Weiergewan,
11, rue Edmond Reuter,
L-5326 Contern, LUXEMBOURG

Office phone :  +352-2454-2566
Office fax:       +352-2454-3566
mobile phone:  +49 151 52 40 66 44

e-mail: alexei.boulbitch at iee.lu

```

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