Integration Involves Tanh, Legendre Polynomial and Gamma function

integration

Solution 1:

I was trying to answer this on Physics Stack exchange and it was migrated as I did so!

This integral is an inverse Mehler transform where the Gamma function and the Tanh occur naturally as poart of the normalization of the *conical functions $P_{-1/2-i\kappa}(x)$, $x\in [1,\infty]$. A Table of Mehler trabsforms appear the Bateman MS project "Tables of Integral Transforms" (Erdelyi et al eds.)

Some notes on these functions and transforms appear in my online notes http://people.physics.illinois.edu/stone/hyperbolic_laplacian.pdf about page 8 or so.

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