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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 R_{1} and R_{2}
shown in Fig. 2. Denote the two segments of C by C_{1} and C_{2}. Applying Cauchy’s theorem we get

Thus

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