The Argument Principle

Let f be a complex-valued, holomorphic function inside and on a simple closed curve C around the origin. The winding number of f about the origin is equal to the number of zeros of f inside C.

Example 1.

According to the Argument Principle, the function f(z) = z2 + 1/2*z will wind around the origin two times as z travels around the unit circle since 0 and -1/2, the two roots of f(z), are both on the unit disk.

Example 2.

The function f(z) = z3 - z - 1 has only two roots in the unit disc so f(z) will wind around the origin two times as z travels around the unit circle.