Rouche's Theorem

Let f(z) and g(z) be complex-valued, holomorphic functions on and inside a simple closed curve C around the origin. If |g(z)| < |f(z)| on C, then f(z) and f(z)+g(z) have the same number of zeros inside C.

Rouche's Theorem is a direct consequence of the Argument Principle and the Dog-on-a-Leash Principle.

Example 1.

Let f(z) = z2 + 1/2*z and g(z) = 1/4*z where z = (cos(t) , sin(t)), 0 <= t <= 2*pi, traverses the unit circle . It is easy to verify that |g(z)| = 1/4 < 1/2 <= |f(z)| for all z on the unit circle. By the Dog-on-a-Leash Principle, f(z) = z2 + 1/2*z and f(z)+g(z) = z2 + 3/4*z wind around the origin twice as z traverses the unit circle once, and by Rouche's Theorem, f(z) and f(z)+g(z) each have two roots with modulus less than one.

Example 2.

Let f(z) = z3 and g(z) = 1/2*z as z traverses the unit circle once. Since we know that f(z) has three zeros at z = 0 (counting multiplicities), we can conclude that f(z) and f(z)+g(z) will wind around the origin three times.