Hi, I think something is wrong with my working because I have a 2 where it should only be a 1.

Q: Show

$\displaystyle \sum_{n=-\infty}^\infty \frac{1}{a^2n^2+1} = \frac{\pi}{a} coth \frac{\pi}{a}$

Working:

I have got the fourier transform of $\displaystyle \frac{1}{a^2n^2+1}$ which is $\displaystyle f(k)=\frac{\pi}{a}e^{-2|k| \frac{\pi}{a}}$.

So from here I use:

$\displaystyle \sum_{m=-\infty}^\infty f(m) = \sum_{n=-\infty}^\infty F(2 \pi n)$

Thus:

$\displaystyle \sum_{n=0}^\infty F(-2 \pi n) + \sum_{n=1}^\infty F(2 \pi n)$ $\displaystyle = \frac{\pi}{a} [2 \sum_{n=1}^\infty e^{-4n\frac{\pi}{a}} -1]$ $\displaystyle = \frac{\pi}{a}[\frac{2}{1-e^{-4\frac{\pi}{a}}} -1]$ $\displaystyle =\frac{\pi}{a} coth \frac{2\pi}{a}$

So where did I go wrong to get the 2 instead of a 1?

Thanks