Evaluation of Improper Integral via Complex Analysis

We are following the notation in Complex Variables and Applications (Brown and Churchill).

The method of using complex analysis to evaluate integrals is to consider a very large semicircular region’s boundary, which consists of the segment of the real axis from $z=-R$ to $z=R$ and the top half of the circle $|z|=R$ positively oriented is denoted by $C_R$ .

$\int_{-R}^R f(x)\,dx+\int_{C_R}f(z)\,dz=2\pi i\sum_{k=1}^n\text{Res}_{z=z_k}f(z)$ . If $\lim_{R\to\infty}\int_{C_R}f(z)\,dz=0$ , then $P.V.\int_{-\infty}^\infty f(x)\,dx=2\pi i\sum_{k=1}^n\text{Res}_{z=z_k}f(z)$ . Furthermore if $f$ is even, then $\int_0^\infty f(x)\,dx=\pi i\sum_{k=1}^n\text{Res}_{z=z_k}f(z)$ .

Useful Theorem

Let two functions $p$ and $q$ be analytic at a point $z_0$ . If $p(z_0)\neq 0, q(z_0)=0$ , and $q'(z_0)\neq 0$ , then $z_0$ is a simple pole of the quotient $p(z)/q(z)$ and $\text{Res}_{z=z_0}\frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}$ .