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.