Prove: Let a function f(z) be analytic in a simply-connected region R and let C be a closed (not necessarily simple) curve in R. Then
See Fig. 1.
Proof. Suppose the curve is curve C shown in Fig. 1. Then C can be viewed as the boundary of two simply-connected region R1 and R2 shown in Fig. 2. Denote the two segments of C by C1 and C2. Applying Cauchy’s theorem we get