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Topic: Help with identity
Replies: 15   Last Post: May 9, 2013 7:13 AM

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Mike Trainor

Posts: 28
Registered: 4/21/13
Re: Help with identity
Posted: Apr 26, 2013 8:10 AM
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On Thu, 25 Apr 2013 09:56:47 -0500, dullrich@sprynet.com wrote:

>On Thu, 25 Apr 2013 08:17:47 -0400, Mike Trainor
><mtrainor@hotmail.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!

mt



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