complex integrals

According to Churchill’s book Complex Variables and Applications,

Integrals are extremely important in the study of functions of a complex variable. The theory of integration, to be developed in this chapter, is noted for its mathematical elegance. The theorems are generally concise and powerful, and many of the proofs are short.

Basic Contour Integrals

The basic way of computing contour integrals is to use the definition. There are more advanced and very powerful methods of computing contour integrals, which we will mention in later posts.

The summarised definition is as follows: $\int_C f(z)\ dz=\int_a^b f[z(t)]z'(t)\,dt$ where $z=z(t), a\leq t\leq b$ represents a contour $C$ .

Basic Example 1: $I=\int_C \bar{z}\,dz$ , where $C$ is the contour $z=2e^{i\theta}$ , $-\pi/2\leq\theta\leq\pi/2$ .

Using the definition, we have

$\begin{aligned} I&=\int_{-\pi/2}^{\pi/2}2e^{-it}\cdot 2ie^{it}\,dt\\ &=4i\int_{-\pi/2}^{\pi/2} 1\,dt\\ &=4\pi i \end{aligned}$

Tag: complex integrals

Basic Contour Integrals