A power series with non-negative power terms is called a Taylor series. In complex variable theory, it is common to work with power series with both positive and negative power terms. This type of power series is called a Laurent series. The primary goal of this chapter is to establish the relation between convergent power series and analytic functions. More precisely, we try to understand how the region of convergence of a Taylor series or a Laurent series is related to the domain of analyticity of an analytic function. The knowledge of Taylor and Laurent series expansion is linked with more advanced topics, like the classification of singularities of complex functions, residue calculus, analytic continuation, etc.
This chapter starts with the definitions of convergence of complex sequences and series. Many of the definitions and theorems for complex sequences and series are inferred from their counterparts in real variable calculus.
Complex sequences and series
An infinite sequence of complex numbers, denoted by {zn}, can be considered as a function defined on a set of positive integers into the unextended complex plane. In other words, the sequence of complex numbers z1, z2, z3, … is arranged sequentially and defined by some specific rule.