# Justify an equality involving hyperbolic cosine and Fourier series

## 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.

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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.

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.