An asymptotic expansion relevant to the summation of csc

It can be shown that 

Consider following complex integral

Where Gamma  is rectangle: ,

use small circular arc with radius  to bypass  and 

Decompose above complex integral into three parts

Where C1 and C2 are both small circular arc

Decompose the first integrand into two parts, expand the second and the third Integrands into series of x

expand csch into series of x, and work out other three integrals

According to this article: https://zhuanlan.zhihu.com/p/430027389

Take the limit as  tend to .