We know $$\sum_{n=1}^\infty\frac{1}{n^2}$$ can be evalute by use Fourier series of $x^2$ on $(-\pi,\pi)$,
if $$\sum_{n=1}^\infty \frac{a}{n^2+a^2}$$ can be evalute use similar approach?
I try to use the poisson summation formula: $$\sum_{n\in Z}f(n)=\sum_{n\in Z}\widehat{f}(n)$$,$f=\frac{a}{x^2+a^2}$, as $f$ is a even function,but I don't know if it is useful.
Thank you for sharing your mind.
Firstly, we have the Fourier series for the hyperbolic cotangent (link):
$$\coth x = \frac{1}{x} + 2 \sum_{n=1}^\infty \frac{x}{x^2+\pi^2n^2}$$
At $x = a\pi$ in particular, with a little shuffling and algebra,
$$\coth(a\pi) = \frac{1}{a\pi} + \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{ a}{a^2 + n^2 }$$
so
$$\sum_{n=1}^\infty \frac{a}{a^2 + n^2} = \frac \pi 2 \coth(a\pi) - \frac{1}{2a}$$
