Laurent Series. Residue Integration
The main purpose of this chapter is to learn about another powerful method for evaluating complex integrals and certain real integrals. It is called residue integration. Recall that the first method of evaluating complex integrals consisted of directly applying Cauchy's integral formula of Sec. 14.3. Then we learned about Taylor series (Chap. 15) and will now generalize Taylor series. The beauty of residue integration, the second method of integration, is that it brings together a lot of the previous material.
Laurent series generalize Taylor series. Indeed, whereas a Taylor series has positive integer powers (and a constant term) and converges in a disk, a Laurent series (Sec. 16.1) is a series of positive and negative integer powers of z − z0 and converges in an annulus (a circular ring) with center z0. Hence, by a Laurent series, we can represent a given function f(z) that is analytic in an annulus and may have singularities outside the ring as well as in the “hole” of the annulus.
We know that for a given function the Taylor series with a given center z0 is unique. We shall see that, in contrast, a function f(z) can have several Laurent series with the same center z0 and valid in several concentric annuli. The most important of these series is the one that converges for 0 < |z − z0| < R, that is, everywhere near the center z0 except ...