Engineering Mathematics, Volume 2

16

Complex Power Series

16.1 Introduction

In calculus, we have studied¹Taylor's and²Maclaurin's power series expansions in the forms Σa_n(z – a)ⁿ and Σa_nzⁿ 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 (z – a). 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, … u_n, … are called the first, the ...