Engineering Mathematics, Volume III

5

Complex Power Series

5.1 Introduction

In calculus, we have studied ¹Taylor’s and ²Maclaurin’s power series expansions in the forms ∑a_n(z − a)ⁿ and ∑a_nzⁿ in respect of elementary functions. We now extend these to functions of complex variables. Further, we study here another type of expansion known as ³Laurent’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.

5.2 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 ...