# Thread: Integral on the Complex Plane

1. ## Integral on the Complex Plane

At the end of my complex analysis class, there is an integral which I can't work out. The main problem I have is the appearance of a singularity on my contours. Does anyone know how to solve it. Unfortunately I've never learnt latex so here it is in word format.

I = -∞Int∞ (cos(x) - 1)/(x^2) dx

Thanks for any help or anyone who can convert that to latex.

2. Your integral can be written as follows:

$\displaystyle \displaystyle{\int_{-\infty}^{\infty}\frac{\cos(x)-1}{x^{2}}\,dx.}$

You can double-click the integral to see the LaTeX code that generated it.

I agree that you can't integrate directly over a pole. What you're going to have to do is skirt the pole by doing an $\displaystyle \epsilon$ semicircle around the pole. It's a first-order pole, by the way, since the numerator has a first-order zero at zero. Once you've computed the integral over the $\displaystyle \epsilon$ semicircle, you can add to that the other two integrals from $\displaystyle (-\infty,-\epsilon)$ and $\displaystyle (\epsilon,\infty).$ Make sense?