## 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) = z^{2} + 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) = z^{3} - 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.