Originally Posted by

**Ackbeet** You can do Green's Theorem. So, in reviewing your work, I'd say you're good up to here:

$\displaystyle \displaystyle{\frac{1}{|z-z_{0}|^{2}}\left[\oint_{C}[(x-x_{0})dx+(y-y_{0})dy]+i\oint_{C}[-(y-y_{0})dx+(x-x_{0})dy)]\right]}. $

Now, what you want to do is apply Green's Theorem on each integral. Green's theorem is going to reduce your integrals quite a bit. Recall that Green's Theorem states that

$\displaystyle \displaystyle{\oint_{C}(L\,dx+M\,dy)=\iint_{D}\lef t(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dx\,dy}.$

Once you have constructed your double integrals, I'd recommend converting to polar coordinates. The result pops out fairly readily.