ABSTRACT
This chapter re-examines complex integration from the viewpoint of the representation of complex functions in terms of series. It shows how, by extending the concept of a Taylor series expansion of a function about a point z0, a Laurent series expansion arises involving a sum of both positive and negative powers of (z z0), where a term of the form (z z0)
n is called a pole of order n. Laurent series expansions describe functions that are analytic at all points of a domain with the exception of certain points, called singularities, where they cease to be differentiable. Laurent series, in turn, lead to the residue theorem that both simplifies the task of evaluating definite integrals and, in particular, makes it possible to evaluate definite integral that are too difficult to determine directly by using only the Cauchy integral theorem, as shown in Chapter 2.
