Help with identity

[Mike Trainor]

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

[dull...@sprynet.com]

On Thu, 25 Apr 2013 08:17:47 -0400, Mike Trainor
<mtra...@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)) ?

>= 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.

??? If you can do that then you've proved the identity;
what's the problem?

[Mike Trainor]

On Thu, 25 Apr 2013 09:56:47 -0500, dull...@sprynet.com wrote:

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

[Robin Chapman]

On 26/04/2013 13:10, Mike Trainor wrote:

> 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.

As David said, summing the claimed answer to give
this function *is* a 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.

How about integrating z^{n-1}/(cosh(x) - (z+1/z)/2)
over the unit circle in C?

[dull...@sprynet.com]

On Fri, 26 Apr 2013 08:10:51 -0400, Mike Trainor

<mtra...@hotmail.com> wrote:

>On Thu, 25 Apr 2013 09:56:47 -0500, dull...@sprynet.com wrote:
>
>>On Thu, 25 Apr 2013 08:17:47 -0400, Mike Trainor
>><mtra...@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).

[Mike Trainor]

On Fri, 26 Apr 2013 09:05:01 -0500, dull...@sprynet.com wrote:

>>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?

That would mean what the word "assigned" means.
If it means homework or an exam, I am 35 years
too old for it. If it means analytically solving a certain fluid
flow problem for an industrial application, then it
it would be ok to call it an assignment. The key
is to understand .... not get answers ... or so
they said all those years ago in graduate school.

mt

[Mike Trainor]

Thanks, Robin, from bringing back 25+ year old
memories ... have not done this kind of work in
a while. Funny about the cosh(x) terms as it
simplifies the terms.

I do have a question as my memory is shot and
I cannot figure it out. I see why you would have
z^(n-1) and not z^n as the 'dy' becomes
dz/(i*z). Now, there are simple poles at
z = exp(+/- x), and only the z = exp(-x) lies
within the contour as x > 0 in my case, at least.
That gives the cosech(x) term I need in the
answer.

But, I have a question. What above the
z^n term in the numberator that comes due
to the numerator that should be there from
the cos(ny) part? That messes up things as
the residues now have exp(-n*x) and,
unforturnately, exp(n*x). Other than that,
it all works out.

I have yet to get to the basement, locate and
pull out my grad school textbooks .... I guess
the answer is there.

Thanks for the pointer and would appreciate it
if you can give me a hint.

tia
mt

[Mike Trainor]

And, I might as well add that I have not gotten
near the n-th order pole at z = 0 :-)

mt

[Robin Chapman]

No we don't. Put z = cos(y) + i sin(y) = exp(iy).
What is z^n in terms of y?

[Robin Chapman]

On 02/05/2013 01:50, Mike Trainor wrote:

>>>
>>> How about integrating z^{n-1}/(cosh(x) - (z+1/z)/2)
>>> over the unit circle in C?

> And, I might as well add that I have not gotten
> near the n-th order pole at z = 0 :-)

The above function hasn't a pole at 0.

[Mike Trainor]

On Thu, 02 May 2013 09:41:43 +0100, Robin Chapman

Yes, I see that -- and we can take the real part.
This also avoids a pole at z = 0.

I was writing cos(n x) as (z^n + z^(-n))/2, just
as in the denominator. I see we do not *have*
to.

Thanks a million .... I think I have got it, but I
will pull out the books and refresh my
knowledge as I am sure there is something
like Jordan's lemma that says why the z^(-n)
term is bad news. The n-th order pole is
definitely bad news :-)

thanks again
mt

[Ray Vickson]

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.

[Mike Trainor]

Finally dug out the books and saw an example
of how to do these integrals -- suddenly
everything came back. The book had a example
of how to do the integral of

cos(3*x)/(a - cos(x))

They did have z^3 + z^(-3) in the denominator
and messed with the 3rd order pole. But, in the
margin, in my own hand was a note saying that
z^3 is enough and that the residues add up to
the same either way!!

Irony (and atrophy), I guess.

Thanks again.

mt

[AP]

On Thu, 25 Apr 2013 08:17:47 -0400, Mike Trainor
<mtra...@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) = 1 +
>
> sum{n, 1, inf} [exp(-nx)*cos(ny)]

not
1+2sum{n, 1, inf} [exp(-nx)*cos(ny)]
?

[Virgil]

In article <v4sjo8hvdpgibvjlq...@4ax.com>,

AP <marc.pi...@wanadoo.fr.invalid> 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)]

sinh(x)/(cosh(x) - cos(y) =

should be

sinh(x)/(cosh(x) - cos(y) ) =
--

[Mike Trainor]

On Wed, 08 May 2013 08:29:45 +0200, AP
<marc.pi...@wanadoo.fr.invalid> wrote:

>On Thu, 25 Apr 2013 08:17:47 -0400, Mike Trainor
><mtra...@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) = 1 +
>>
>> sum{n, 1, inf} [exp(-nx)*cos(ny)]
>not
>1+2sum{n, 1, inf} [exp(-nx)*cos(ny)]
>?

You are right. Antoher typo!

mt