I'm familiar with Cauchy's residue theorem for calculating integrals in the complex plane. I'm wondering if there's a natural way of extending this to functions which also contain branch cuts. As an example consider $$ \int_{|z|=1}\sqrt{(2+z)(2+1/z)}dz $$ This has a pole of order $-\frac{1}{2}$ at the origin with a zero of order $\frac{1}{2}$ at $z=-\frac{1}{2}$ and a branch cut joining the two. How can one evaluate this exactly using contour integration? Is there some sort of combined residue of the whole cut?
Thanks in advance for any help.