Prove. Let a function f(z) be analytic with a continuous derivative f '(z) within and on the boundary of a region R, either simply or multiply-connected, and let C be the entire boundary of R. Then
Proof. We first prove the theorem for a simply-connected region. Let the region R be simply-connected. Let f(z) = u(x, y) + i v(x, y). Then the integral around the boundary C is given by
Since f(z) is analytic with a continuous derivative in R and on the boundary C it follows that the conditions for Green’s theorem are met. Applying Green’s theorem to the two integrals in the right member we get
Because f(z) is analytic the Cauchy-Riemann equations
are satisfied and we see that both integrands in the right member of 2) are zero. Thus
Note. We have given the proof with the added condition that f '(z) be continuous in R but a proof can be given without this condition. See Spiegel. Complex Variables. (Schaum) Chap. 4.
Proof for multiply-connected regions: Consider the multiply-connected region shown in Fig. 1 where cross-cuts have been constructed connecting the interior and exterior boundaries thus transforming the multiply-connected region into a simply-connected one where Cauchy’s theorem for simply-connected regions applies. We let the curve C of the theorem be the boundary of the simply-connected region shown in Fig. 1. The theorem states that
The total amount of curve traversed is equal to the boundaries C, C1, C2, C3, C4 and C5 plus all the cross-cuts traversed. However, each cross-cut is traversed in opposite directions and the line integrals on the cross-cuts cancel each other out so the net amount of curve traversed that counts in the total is simply the boundaries C, C1, C2, C3, C4 and C5 . And this is what Cauchy’s theorem of multiply-connected regions states.
Spiegel. Complex Variables. (Schaum) Chap. 4.