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

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David C. Ullrich

Posts: 3,084
Registered: 12/13/04
Re: Help with identity
Posted: Apr 26, 2013 10:05 AM
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On Fri, 26 Apr 2013 08:10:51 -0400, Mike Trainor
<mtrainor@hotmail.com> wrote:

>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.


That's "the" problem, right. Does that mean it's
like an assigned problem, or just something you're
wondering about?

If the latter, you should consider that it does
happen sometimes that the only, or the only
reasonable, way to show that f^(n) = c_n is
to show that f(t) = sum c_n exp(int).



>
>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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