Let be a closed path in a domain such that for all . Suppose that is analytic on except possibly at a finite number of isolated singularities . Show that .
Consider the Laurent decomposition at each .
I do not see how to get the Laurent decomposition of the to do this problem. By the way, the above notation means the winding number. It looks like perhaps the residue theorem would come into play here somehow too. I just do not see how to proceed now. Any suggestions would be very nice. Thank you.