# Complex Fourier Series Problem

1. Apr 14, 2008

### White Ink

1. The problem statement, all variables and given/known data

Having shown that:

$$f(x) = e^{ax} = \frac{sinh(a\pi)}{\pi}\sum^{\infty}_{n=-\infty}\frac{(-1)^{n}(a+in)}{(a^{2}+n^{2})}e^{inx}$$

on $$(-\pi,\pi)$$ with 'a' a real constant

Deduce that:

$$\pi coth(a\pi) = \frac{1}{a} + \sum^{\infty}_{n=1}\frac{2a}{(a^{2}+n^{2})}$$

2. Relevant equations

$$sinhx = \frac{e^{ax} - e^{-ax}}{2}$$

$$coshx = \frac{e^{ax} + e^{-ax}}{2}$$

$$cothx = \frac{coshx}{sinhx}$$

3. The attempt at a solution

I was able to derive the very top equation; I then set $$x=\pi$$ (in the top equation) so that it reduced to:

$$e^{a\pi} = \frac{sinh(a\pi)}{\pi}\sum^{\infty}_{n=-\infty}\frac{(a+in)}{(a^{2}+n^{2})}e^{inx}$$

I know that by fiddling around with the definition of cothx I can get fairly close to what I want, but my major issue here is the change of the 'interval' of summation between what I have shown and what I am aiming to show. Any help would be greatly appreciated.

Last edited: Apr 14, 2008
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