Applied Mathematical Methods

47

Singularities of Complex Functions

In the first chapter on complex analysis (Chap. 45), we concentrated on analytic functions. In the next, we witnessed the interplay of analyticity and singularity, operating side by side. Finally, in the present chapter, our major focus is on singularities. Here, we apply the results of the previous chapter to develop series representations of complex functions. Next, in the framework of Laurent’s series, the idea of singularities is developed, leading to the theory of residues.

Series Representations of Complex Functions

If function f (z) is analytic in a neighbourhood of the point z₀, then f (z) has a unique power series representation in powers of (z - z₀), known as Taylor’s series, given by

the coefficients ...