Homework Help: Integral using CIF

1. Oct 8, 2011

Mechdude

1. The problem statement, all variables and given/known data
evaluate
$$\int_{-\infty}^{\infty} \frac{dx}{x^2 - \sigma^2}$$

2. Relevant equations

CIF: $$\int f(z) = 2 \pi i \times \sum Res f(z) |_a$$

3. The attempt at a solution
let
$$\int_c \frac {dz}{z^2 - \sigma^2} = \int_c \frac{1}{(z - \sigma)(z+ \sigma)}$$
with poles at $z = \pm \sigma$ but i take only those in the upper half plane, and thus the residue:
$$\lim_{ z \to \sigma} (z-\sigma) \frac{1}{( z- \sigma)( z + \sigma)} = \frac{1}{\sigma + \sigma} = \frac{1}{2 \sigma}$$
with the integral becoming :
$$2 \pi i \sum Res f(z) = 2 \pi i \frac{1}{2 \sigma} = \frac{\pi i }{\sigma}$$
1. let $\sigma \to \sigma +i \gamma$
2. let $\sigma \to \sigma - i \gamma$