A Fourier series arising in perturbation theory in quantum mechanics is
$$\sum_{m\neq n} \frac{1}{n^2 - m^2} \cos \frac{m\pi x}{2a} \, .$$
where $n$ is an odd positive integer and $n$ runs through all odd positive integers other than $n$. (The numbers are odd so that the Fourier terms are zero at $\pm a$.)
I have no idea what kind of function produces this series. Is it familiar to anyone?