# 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 $\text{Res}_{z=z_0}\frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}$. ## Author: mathtuition88

http://mathtuition88.com

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