Advanced Calculus fi..

Chapter 8 Functions of a Complex VariableIf this series converges for lzl I c r;, then its sum is an analytic function F(z1): hencethe series (8.58) converges forto the analytic function g(z) = F(l/(z - 20)). The value zl = 0 corresponds toz = oo, in a limiting sense; accordingly, we can also say that g(z) is analytic at ooand g(oo) = 0. This will be justified more fully in Section 8.14.The domain of convergence of the series (8.58) is the region (8.59), which is theexterior of a circle. It can happen that r; = oo, in which case the series convergesfor all z except zo; if r; = 0, the series diverges for all z (except z = oo, as above).If we add to a series (8.58) a usual power series2 an(z - ZO)" = a0 + al(z - ZO) + ...,n=Oconverging for (z- zol < r;, we obtain a Sum .If r,* < r;, the sum converges and represents an analytic function f (z) in the annulardomain: r; < lz - zol < r;, for each series has an analytic sum in this domain, sothat the sum of the two series is analytic there. We can write this sum in the morecompact form (after some relabeling)though this should be interpreted as the sum of two series, as in (8.60).In this way we build up a new class of analytic functions, each defined in aring-shaped domain. Every function analytic in such a domain can be obtained inthis way:THEOREM 25Then(Laurent's Theorem) Let f (z) be analytic in the ring:RI < - zol < R2.whereand C is any simple closed curve separating lzl = RI from lzl = R2. The seriesconverges uniformly for RI < k~ 5 lz - zol 5 k2 R2.