In evaluating the line integral of $\frac{dz}{z-2}$ around the circle $|z-1|=5$ , and also around the square with vertices $3+3i ,3-3i,-3+3i,-3-3i$, I obtain zero in both cases however my textbook gives $2\pi i$ in both cases. Am I correct here?
(The usual parametrizations obviously work here.....)
Integral is sum of residues at poles multiplied by $2\pi i$ , here $z=2$ is a simple pole and residue at $z=2$ is 1.