
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
$\displaystyle \sum_{n=\infty}^\infty \frac{1}{a^2n^2+1} = \frac{\pi}{a} coth \frac{\pi}{a}$
Working:
I have got the fourier transform of $\displaystyle \frac{1}{a^2n^2+1}$ which is $\displaystyle f(k)=\frac{\pi}{a}e^{2k \frac{\pi}{a}}$.
So from here I use:
$\displaystyle \sum_{m=\infty}^\infty f(m) = \sum_{n=\infty}^\infty F(2 \pi n)$
Thus:
$\displaystyle \sum_{n=0}^\infty F(2 \pi n) + \sum_{n=1}^\infty F(2 \pi n)$ $\displaystyle = \frac{\pi}{a} [2 \sum_{n=1}^\infty e^{4n\frac{\pi}{a}} 1]$ $\displaystyle = \frac{\pi}{a}[\frac{2}{1e^{4\frac{\pi}{a}}} 1]$ $\displaystyle =\frac{\pi}{a} coth \frac{2\pi}{a}$
So where did I go wrong to get the 2 instead of a 1?
Thanks

Re: Poisson summation
I know $\displaystyle coth \frac{\pi}{a}= \frac{1+e^{2\frac{\pi}{a}}}{1e^{2\frac{\pi}{a}}}$, but I have a 4 where the 2 is, and I don't know where I went wrong.