The problem is calculating $$ \oint_C(z-z_0)^m dz , m=0,\pm1,\pm2,...$$ Where C is any simple closed path enclosed z0, counterclockwise.
So I know that solving by using K which is the circle $$ |z-z_0|=ρ ,counterclockwise$$ when m is non-negative integer, since $$ |z-z_0|=ρ$$ is entire function and by the Cauchy integral Thm, $$ \oint_K(z-z_0)^m dz=0$$
but I don't know how to solve this when m is negative integer.
The answer is $$\begin{cases} 2\pi i \: (m = −1) \\ 0 \: (m \ne −1) \end{cases} $$