zero as contour integral

Suppose that f is a complex function which is defined in some open set D which has a simple zero at some point pD. Then we have


where C is a closed path in D which encloses p but does not enclose or pass through any other zeros of f.

This follows from the Cauchy residue theorem. We have that the poles of f/f occur at the zeros of f and that the residueMathworldPlanetmath of a pole of f/f is 1 at a simple zero of f. Hence, the residue of zf(z)/f(z) at p is p, so the above follows from the residue theorem.

Title zero as contour integral
Canonical name ZeroAsContourIntegral
Date of creation 2013-03-22 16:46:42
Last modified on 2013-03-22 16:46:42
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Corollary
Classification msc 30E20