Our next few posts on complex analysis will focus on evaluating real integrals like 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 be a simple closed positively oriented contour. If a function is analytic inside and on except for a finite number of singular points () inside , then .
A Summary of the 3 types of Isolated Singular Points:
- Pole of order . The coefficients of the Laurent series contain a finite (nonzero) number of nonzero terms, i.e. eventually becomes zero after a certain number. i.e. .
- Removable singular point. Every is zero.
- Essential singular point. An infinite number of the coefficients in the principal part are nonzero.
Shortcut for calculating Residues at Poles
Theorem: An isolated singular point of a function is a pole of order if and only if can be written in the form where is analytic and nonzero at . Moreover if and . if .
This is a very short crash course on the theorems needed. The next blog post on complex analysis will go into calculating some actual integrals.