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Last week I tried to find a proof for the so called partial fraction decomposition of the cotangent, that is

$$\pi \, \cot(\pi k) = \sum_{m=-\infty}^{\infty} \frac{k}{k^2-m^2}$$

For the proof I used Fourier-series in the following way:

An even function $f(w)=f(-w)$ fulfills

$$2 \pi f(z) = \sum_{m=-\infty}^\infty \cos(m z) \left( \int_{-\pi}^\pi f(w) \cos(m w) dw \right)$$

Now I take the even function $f(z) = \cos(k z)$ and get for the Fourier-coefficients:

$$a_m = \int_{-\pi}^{\pi} \cos(m w) \cos(k w) dw = \frac{2 \sin(\pi k) (-1)^m k}{k^2-m^2}$$

It follows, that

$$2 \pi \cos(k z) = \sum_{m=-\infty}^\infty \frac{2 \sin(\pi k) (-1)^m k \cos(m z)}{k^2-m^2}$$

Putting $z = \pi$ and using $\cos(m \pi) = (-1)^m$ gives the sought for formula.

This seems to be more than a special ad-hoc trick, but a method to compute the sum of a lot of series by fourier expansion of suitable "kernels" (like $\cos(k z)$ above):

To sum $\sum_{m=-\infty}^\infty g(z,m)$ find a kernel $f(t,z)$ and a fourier expansion (for simplicity in formulas I consider $f(t,z)$ even in $t$):

$$2 \pi f(t,z) = \sum_{m=-\infty}^\infty g(z,m) h(z) \cos(m t)$$

so that

$$\int_{-\pi}^\pi f(w,z) \cos(m w) dw = g(z,m) h(z)$$

holds. Then

$$\sum_{m=-\infty}^\infty g(z,m) \cos(m t_0) = 2 \pi f(t_0,z)/h(z) $$

Now my question is: To compile in the answers a list of applications of the principle above to different examples from "practice".

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    $\begingroup$ You should look at the complex analysis method to arrive at those results, in particular that $\frac{\pi^2}{\sin^2(\pi z)}-\sum_n \frac1{(z-n)^2}$ is a $1$-periodic entire function, bounded on $\Re(z)\in [0,1]$, thus bounded everywhere, thus it is constant, and since it vanishes at $i\infty$, it is $=0$. With the same kind of argument we get a pole expansion of most quotients of trigonometric functions, as well as their primitives and their multiplication with rational functions, from which we get a closed-form for a lot of series. This is more or less the big list you are asking for. $\endgroup$
    – reuns
    Jan 20, 2020 at 13:03

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