I am here again! This time is a complex integral question! I can not work it out by myself!
Consider
$I(x)=\sum_{n=-\infty}^{+\infty}g_n(x)$ and $g_n(x)=\frac{1}{i(2n+1)-x}$, where x is a real variable and i is the imaginary unit.
The goal of this question is to calculate the sum above by transforming it into a complex integral. The hint is to show that $I(x)=\int_Cf(z)\frac1{z-x}dz$ by introducing z as a complex number and $f(z)=\frac1{2\pi i}(\frac1{e^z+1})$ and choose a proper contour C for integration at the same time. Then calculate the integral to get $I(x)$
From my point, there must be a connection between the singularities of $f(z)$ and the $i(2n+1)\pi$ in $g_n(x)$ and I need to use the residue. Then, game is over.
Best Answer
Proof sketch that uses the fact that $f(z) + f(-z) = 1$: