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

.

Hint

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.