August 15, 2012 1 Comment
Today we prove Rouché’s theorem. The gist is that it helps us count the number of roots of a holomorphic function, given some bounds on its values.
Theorem 1 Suppose and are holomorphic functions inside and on the boundary of some closed contour . If
on , then and have the same number of zeros on the interior of .
Before we begin proving this, it should be emphasized that we count with multiplicity. We would count the number 1 as a root of twice. Most root counting we every do will be done this way. I feel confident in saying that it is the correct way to count roots, even if at first it is unintuitive.
Proof: By hypothesis, has no roots on the boundary . Define . The roots of are the roots of . The poles of are the roots of . So it suffices to use the argument principal to show that . That is, we need to show that
But from our hypotheses, we can conclude that
That is, never takes on values more than away from . Imagine a dog tethered by a leash of length less than to the point . That dog can’s reach the origin, but more importantly, he can’t walk a loop around the origin. So computing the winding number about the origin must give us zero. This proves the theorem.