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## Complex Power Series

#### 16.1 Introduction

In calculus, we have studied1Taylor's and2Maclaurin's power series expansions in the forms Σan(za)n and Σanzn in respect of elementary functions. We now extend these to functions of complex variables. Further, we study here another type of expansion known as3Laurent's series expansion which involves both positive as well as negative powers of (za). These expansions help us in the study of the properties of functions, in numerical computations and in the evaluation of certain real integrals.

#### 16.2 Sequences and Series

Let S be a set of complex numbers. We can form a complex sequence by choosing from S a first number u1, a second number u2, and so on. Here u1, u2, … un, … are called the first, the ...

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