An asymptotic expansion relevant to the summation of csc
2022-10-10 18:38 作者:Baobhan_Sith | 我要投稿
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
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