1
vote
1answer
77 views

Calculate the residue of $\cot\pi z$ at poles $z=n$

I'm having trouble calculating the residue of $f(z) =\cot\pi z$. The function has a simple pole for every integer n, and i'm, trying to find the residue at n. I know that by the residue theorem: ...
4
votes
1answer
129 views

Integral using residue theorem (maybe)

I came across the following integral in a book (Kato's Perturbation Theory for Linear Operators, $\S$3.5): $\int_{-\infty}^\infty (a^2+x^2)^{-n/2}\,dx$ where $n$ is a non-negative integer and $a$ is ...
10
votes
1answer
997 views

Integral Representation of Infinite series

Let's take a look at the following integrals : 1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -\frac 1 2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= -\frac 1 2 ...