# Poisson summation

• October 23rd 2011, 05:53 AM
ellenu485
Poisson summation
Hi, I think something is wrong with my working because I have a 2 where it should only be a 1.

Q: Show

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

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

So from here I use:

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

Thus:

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

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

Thanks
• October 23rd 2011, 01:28 PM
ellenu485
Re: Poisson summation
I know $coth \frac{\pi}{a}= \frac{1+e^{-2\frac{\pi}{a}}}{1-e^{-2\frac{\pi}{a}}}$, but I have a 4 where the 2 is, and I don't know where I went wrong.