Advanced Calculus fi..

Chapter 8 Functions of a Complex Variable 557Now let D be a simply connected domain and let zo be a fixed point of D. If f (z) isanalytic in D, the function f (z)/(z - zo) will fail to be analytic at zo. Hencewill in general not be zero on a path C enclosing zo. However, as above, this integralwill have the same value on all paths C about zo. To determine this value, we reasonthat if C is a very small circle of radius R about zo, then f (zo) has, by continuity,approximately the constant value f (zo) on the path. This suggests thatsince we findwith the aid of the substitution: z - zo = ~e". The correctness of the conclusionreached is the content of the following fundamental result: ,THEOREM 17 (Cauchy Integral Formula) Let f (z) be analytic in a domainD. Let C be a simple closed curve in D, within which f (z) is analytic and let zo beinside C. ThenProof. The domain D is not required to be simply connected, but since f is analyticwithin C, the theorem concerns only a simply connected part of D, as shown inFig. 8.9. We reason as above to conclude thatFigure 8.9Cauchy integral formula.