# Thread: Complex Contour Integral

1. ## Complex Contour Integral

Let C be any simple closed contour described in the positive sense. Given $g(z)=\int_{C} \frac{s^3+2s}{(s-z)^3}ds$, a)show that $g(z)=6 \pi iz$ 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?

2. Originally Posted by davesface
Let C be any simple closed contour described in the positive sense. Given $g(z)=\int_{C} \frac{s^3+2s}{(s-z)^3}ds$, a)show that $g(z)=6 \pi iz$ 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