Read this: The Cauchy-Goursat Theorem
Let C be any simple closed contour described in the positive sense. Given , a)show that when z is inside C and b)show that g(z)=0 when z is outside.
Part a) was easy, but I don't see how to start part b). The Cauchy integral formula has the requirement that the point of interest be interior to C, so how else could I go about showing g(z)=0 for an arbitrary simple closed contour when the point is exterior to C?
Read this: The Cauchy-Goursat Theorem