Value of $\sum_{k=1}^{\infty}\frac{1}{k^2+a^2}$

Solution 1:

Using the residue theorem, you can show that

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

This is equivalent to saying that

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

You can also derive this by considering the Maclurin expansion of $z \coth{z}$:

$$z \coth{z} = 1 + \sum_{k=1}^{\infty} \frac{B_{2 k} (2 z)^{2 k}}{(2 k)!}$$

where $B_{2 k}$ is a Bernoulli number, which also shows up in Riemann zeta functions of even, positive argument:

$$\zeta(2 k) = (-1)^{k+1} \frac{B_{2 k} (2 \pi)^{2 k}}{2 (2 k)!}$$

To evaluate the sum, factor out $n^2$ from the denominator and Taylor expand:

$$\begin{align}\sum_{n=1}^{\infty} \frac{1}{n^2+a^2} &= \frac{1}{n^2} \frac{1}{1+ \frac{a^2}{n^2}}\\ &= \sum_{n=1}^{\infty} \frac{1}{n^2} \sum_{k=0}^{\infty} (-1)^k \left (\frac{a^2}{n^2}\right )^{k} \\ &=\sum_{k=0}^{\infty} (-1)^k a^{2 k}\sum_{n=1}^{\infty} \frac{1}{n^{2 k+2}} \\ &=\sum_{k=0}^{\infty} (-1)^k a^{2 k} \zeta(2 k+2)\\ &= \frac{1}{2 a^2}\sum_{k=1}^{\infty} \frac{B_{2 k} (2 \pi a)^{2 k}}{(2 k)!} \\ &= \frac{1}{2 a^2} ( \pi a \coth{\pi a} - 1)\\ \end{align}$$

The result follows.

Solution 2:

Look at function $ f(x)=e^{ax},~-\pi\leqslant x\leqslant\pi $. It is known (evaluate) that $ f(x)=e^{ax}, ~-\pi\leqslant x\leqslant\pi $ has the Fourier series

$$ \frac{\sinh\pi a}{\pi a}+\frac{2\sinh\pi a}{\pi}\sum_{n=1}^{\infty}(-1)^n\left[\left(\frac{a}{a^2+n^2}\right)\cos{nx}-\left(\frac{n}{a^2+n^2}\right)\sin nx\right]. $$

The function $f(x)$ is continuous for $ -\pi\leqslant x\leqslant\pi $, but $ f(-\pi)\neq f(\pi) $. Thus, at the ends of the fundamental interval the Fourier series for $f(x)$ will converge to the value $$ \frac{1}{2}\left(f(\pi)+f(-\pi)\right)=\frac{1}{2}\left(e^{a\pi}+e^{-a\pi}\right)=\cosh\pi a. $$

Using this result and setting $x=\pi$ in the Fourier series gives $$\cosh\pi a =\frac{\sinh\pi a}{\pi a}+\frac{2\sinh\pi a}{\pi}\sum_{n=1}^{\infty}\left(\frac{a}{a^2+n^2}\right),$$

where use have been made of the result $ \cos n\pi = (-1)^n $. Thus $$ \coth\pi a =\frac{1}{\pi}\left[\frac{1}{a}+2\sum_{n=1}^{\infty}\frac{a}{a^2+n^2}\right]$$ or, equivalently $$\frac{1}{2}\left(\pi\coth\pi a-\frac{1}{a}\right) =\sum_{n=1}^{\infty}\frac{a}{a^2+n^2}.$$

And now we find $$\sum_{n=1}^{\infty} \frac{1}{n^2+a^2} = \frac{1}{2} \left (\frac{\pi}{a} \coth{\pi a} - \frac{1}{a^2}\right ).$$