Evaluating Integrals using Complex Analysis

Our next few posts on complex analysis will focus on evaluating real integrals like $\displaystyle\int_0^\infty \frac{1}{x^2+1}\,dx$ using residue theory from Complex Analysis. This is something amazing about Complex Analysis, it can be used to solve integrals in real numbers, something which is not immediately obvious.

To calculate those real integrals, the first step is to study the theory of residues and poles. This can be found in Chapter 6 of Churchill’s book Complex Variables and Applications (Brown and Churchill).

The extremely powerful theorem that one first needs to know is called Cauchy’s Residue Theorem:

Let $C$ be a simple closed positively oriented contour. If a function $f$ is analytic inside and on $C$ except for a finite number of singular points $z_k$ ( $k=1,2,\dots,n$ ) inside $C$ , then $\displaystyle \int_Cf(z)\,dz=2\pi i\sum_{k=1}^n\text{Res}_{z=z_k}\,f(z)$ .

A Summary of the 3 types of Isolated Singular Points:

Pole of order $m$ . The coefficients $b_n$ of the Laurent series contain a finite (nonzero) number of nonzero terms, i.e. $b_n$ eventually becomes zero after a certain number. i.e. $\displaystyle f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n+\frac{b_1}{z-z_0}+\frac{b_2}{(z-z_0)^n}+\dots+\frac{b_m}{(z-z_0)^m}$ .
Removable singular point. Every $b_n$ is zero.
Essential singular point. An infinite number of the coefficients $b_n$ in the principal part are nonzero.

Shortcut for calculating Residues at Poles

Theorem: An isolated singular point $z_0$ of a function $f$ is a pole of order $m$ if and only if $f(z)$ can be written in the form $f(z)=\frac{\phi(z)}{(z-z_0)^m}$ where $\phi(z)$ is analytic and nonzero at $z_0$ . Moreover $\text{Res}_{z=z_0}f(z)=\phi(z_0)$ if $m=1$ and $\text{Res}_{z=z_0}f(z)=\frac{\phi^(m-1)(z_0)}{(m-1)!}$ . if $m\geq 2$ .