Argument principle: Encyclopedia II - Argument principle - Consequences

Argument principle - Consequences

This has consequences in considering the winding number of f(z) about the origin, say, if C is a closed contour centered on the origin. We see that the integral of f′(z)/f(z) about C is the change in values of log f(z). Since C is closed we only need consider the change in i arg f(z) over C − which will be some multiple of 2πi since C is closed (but may wind more than once about the origin). But since by the argument principle

the factors of 2πi cancel and so we are left with

where I(C,0) denotes the winding number of f over C about 0.

Adapted from the Wikipedia article "Consequences", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki