On Thu, 25 Apr 2013 09:56:47 -0500, firstname.lastname@example.org wrote:
>On Thu, 25 Apr 2013 08:17:47 -0400, Mike Trainor ><email@example.com> 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) > >Unclear what you mean, since the parentheses are not >balanced. Maybe sinh(x)/(cosh(x) - cos(y)) ?
Sorry, yes. Your guess is correct.
>??? If you can do that then you've proved the identity; >what's the problem?
The problem is to start with the LHS and get the RHS. As I said, this a Fourier series expanison and I want to know how one gets the coefficients.
The problem can be restated as follows:
Find the Fourier series expansion in y of
sinh(x)/(cosh(x) - cos(y)).
As I said in my first note, Bateman *gives* the answer by stating it without proof. I am looking for the proof.
It comes down to doing the integral of
cos(ny)/(cosh(x) - cos(y))
from 0 to 2 pi, for integer n, where x => 0.
Everything is proper for x> 0 (no divergences) and perhaps one can show that x = 0 can be handled. But, I cannot find the integral in the tables and online methods fail. The interesting thing is that the coefficients are so simple!