zero as contour integral
Suppose that is a complex function which is defined in some openset which has a simple zero at some point. Then we have
where is a closed path in which encloses but does not enclose orpass through any other zeros of .
This follows from the Cauchy residue theorem. We have that the poles of occur at the zeros of and that the residue of a pole of is at a simple zero of . Hence, the residue of at is , so the above follows from the residue theorem.