# Thread: Complex Contour Integration With Branch Cut

1. ## Complex Contour Integration With Branch Cut

Hi i'm stuck on a problem and was wondering if anyone could help. The problem is as follows:

"It is required to evaluate
$L=\int_{- \ 1}^{1}\frac {({1-x^2})^\frac{1}{2}}{1+x^2}dx$,
using contour integration.

a) consider the function $f(z)=\frac {({z^2-1})^\frac{1}{2}}{z^2+1}$
(You are given that, with $-\pi<\arg(z\pm1)\leq\pi$, the only branch cut needed is the section [-1,1] of the real axis)

i)Find the value of $M=\oint f(z)dz$ around a circle of large radius.
ii)Relate M to the value of $N=\oint f(z)dz$ around the cut.
iii)Express N in terms of L, and hence evaluate L."