Cauchy-Goursat

If a function f is analytic throughout a simply connected domain D, then

int{C}{}{f(z)dz}=0

for every closed contour C lying in D.

The proof is easy if C is a simple closed contour or if it is a closed contour that intersects itself a finite number of times. For, if C is simple and lies in D, the function f is analytic at each point interior to and on C ; and the Cauchy-Goursat theorem ensures that equation above holds.

The Cauchy-Goursat theorem can also be extended in a way that involves integrals along the boundary of a multiply connected domain. The following theorem is such an extension

(i) C is a simple closed contoul; described in the counterclockwise direction;

(ii) C_k ( k = 1,2, . . . ,n ) are simple closed contours interior to C, all described in
the clockwise direction, that are disjoint and whose interiors have no points in common.

If a function f is analytic on all of these contours and throughout the multiply connected domain consisting of all points inside C and exterior to each C_k,then

int{C}{}{f(z)dz}+sum{k=1}{n}{int{C_k}{}{f(z)dz}}=0