# Homework Help: Residue theorem appliation

1. Mar 2, 2010

### zetafunction

1. The problem statement, all variables and given/known data

Calculate the integral $$I(k)= \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+1)^{k}}$$ with 'k' being a real number

2. Relevant equations the integral equation above

3. The attempt at a solution

from the residue theorem , there is a pole of order one at $$2+ix=0$$ , my problem is the pole of order 'k' at s=i and s=-i , in order to handel with this pole i have thought that using residue theorem

$$\frac{1}{\Gamma(s)}D^{k-1}((s-i)^{k}\frac{1}{(x^{2}+1)^{k}}$$

evaluated at both s=i and s=-i