Re: Integrating special functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg131594] Re: Integrating special functions*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Tue, 10 Sep 2013 03:33:06 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-outx@smc.vnet.net*Delivered-to*: mathgroup-newsendx@smc.vnet.net*References*: <l0etek$20t$1@smc.vnet.net>

Am 07.09.2013 11:59, schrieb Herman: > 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? > I don't want to invest an hour to solve a somehow trivial but a bit tedious problem. So I only will give you some hints. The associated Legrendre function P[n,n,x] is something like (1-x^2)^n for integer n. It represents the extreme form of the angular momentum wave density concentrated at the equator of sphere, rotating wave part e^(i n phi) suppressed, if the argument, the Tanh-function varies in (-1,1). LengendreP[n,n,Tanh[x]] -> constant * Sech[x]^n ;/ n>0 && n\in Integers The product of a function and its derivative is the derivative of its square f'[x] f[x] <-> D[ 1/2 f[x]^2, x] The integral can be integrated by parts because Limit[ x Sech[x]^n], x->+-oo ] ==0 So you problem is only to integrate Integrate[ x D[1/2 Sech[a x]^(2n),x ],{x,-oo,oo}] or - 1/2 Integrate[Sech[a x]^(2n) ,{x,-oo,oo}] -- Roland Franzius