Justify an equality involving hyperbolic cosine and Fourier series

[Emspak]

Homework Statement

The problem:

Justify the following equalities:
$$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$

I am trying to figure out how to start this. When I insert the Euler identity of $\coth$ (using the formula for complex Fourier series) I end up with: $$c_n = \frac{1}{2\pi}\int^{\pi}_{-\pi}\frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}e^{inx} \ dx = \frac{1}{2\pi} \int^{\pi}_{-\pi}\frac{e^{(1+in)x} + e^{-(1+in)x}}{e^{x} - e^{-x}} \ dx$$

which is one ugly integral. So my question is a) did I make a mistake in the starting point and b) can this integral be simplified in some way that's better? Or is there some stupidly silly pattern I should be recognizing here? (I considered treating the integral as $\frac{1}{2\pi} \int^{\pi}_{-\pi} \frac{u}{du}$ or something like it).

I suspect I am missing something obvious.

Thanks for any help. This one I think involves doing out a complex Fourier series for cosh or sinh, which might be simpler. But I am not sure.

 

[brmath]

Here's a thought. Are you familiar with breaking up a fraction like $\frac{A + B} {C + D}$? Just say $\frac {x}{C} + \frac {y}{D}$ = $\frac{A + B} {C + D}$. On the left side, add up the two fractions as you did in elementary school and solve for x and y.

If you do that to your integrand, you will wind up with the sum of two exponentials, and I think you can just integrate it. Whether that trick gets you to the right series I don't know, but it's worth a shot.

I'm also a little worried about your summations -- when you multiply through by i don't you get a -x in the numerator?

Finally -- are you sure this is a Fourier series problem? Is it in that chapter of some book? I am not above expanding cotx in a Taylor's series and seeing if that can be kicked around to get what you need.

 

[Emspak]

It's in the Fourier Series chapter, and the section on complex Fouriers, so presumably they want something like that. There's an earlier problem in the group where they tell you to get a complex Fourier of cosh, and that would be used here, but I am not entirely sure how that would work.