## The Argument Principle

Let be a function analytic on a simply connected region and let be a simple closed contour in such that for Then the winding number of about 0 is given by where is the number of zeros of inside counted according to their orders.

Proof:

It is sufficient to prove that  (using the chain rule) since is a parametrization of this argument is only valid if is non zero on for then the image of each constituent smooth path of is a constituent smooth path of so that is a contour. In general however can have zeros on but only a finite number. By making small detours to near the zeros of we can obtain a simple closed contour on which is never zero and for which

1. encloses the zeros of as 2. Then since we have The principle can easily be extended to a contour about any point %alpha :

Let be a function analytic on a simply connected region and let be a simple closed contour in such that for Then the winding number of about is given by is the number of zeros of inside counted according to their order. 