Date: May 2, 2013 4:06 PM
Author: RGVickson@shaw.ca
Subject: Re: Help with identity
On Thursday, April 25, 2013 5:17:47 AM UTC-7, Mike Trainor wrote:

> Would greatly appreciate any pointers to proving the following

>

> identity that I came across in Bateman's book on partial

>

> differential equations:

>

>

>

> sinh(x)/(cosh(x) - cos(y) = 1 +

>

>

>

> sum{n, 1, inf} [exp(-nx)*cos(ny)]

>

>

>

> It is clearly a simple Fourier expansion in y, but how does

>

> one get the coefficients -- in other words do the integral.

>

>

>

> OTOH, it is trival to prove that the RHS can be summed

>

> to give the LHS.

>

>

>

> The one out I see it expressing either the trignometric

>

> or hyperbolic functions as the other type by using an

>

> imaginary arguments, but I could not get anywhere.

>

>

>

> Also 0<= x < inf and 0 <= y <= 2 pi

>

>

>

> tia

>

> mt

Maybe nobody ever did the integrals to get the series; maybe in the distance past somebody obtained the series, perhaps as part of a larger project, and then asked themselves: hmmmm ... I wonder if the series can be expressed in closed form? Then maybe they discovered the function you have started with.