Math 313 - Functions of a Complex Variable

Demonstration of the Argument Principle

The following animation shows the circle with radius R and its image under the map f(z) = sin(z+1)/z. This function has a simple pole at z=0 (marked with a red x in the plot on the left), and simple zeros at z = n π - 1 (marked with green circles in the plot on the left).

The argument principle says that

Δ_C arg f(z) = 2π(Z-P) where Z and P are the number of zeros and poles, respectively, of f(z) inside C, counted with multiplicity. The integer Z-P is the winding number of f(C); it counts the net number of times the curve winds around the origin.

We apply this principle to f(z)=sin(z+1)/z with C the circle of radius R centered at the origin.

For any R > 0, we have P=1.
For 0 < R < 1, we have Z=0, so the winding number is -1; the curve f(C) goes around the origin once in the clockwise direction.
For 1 < R < π-1, we have Z=1, so the winding number is 0; the curve f(C) does not enclose the origin.
For π-1 < R < 2π-1, we have Z=2, so the winding number is 1; f(c) goes around the origin once in the counterclockwise direction.
More generally, for nπ-1 < R < (n+1)π-1 with n>1, we have Z = n+1, so the winding number of f(C) is n.

In the animated plot below, observe how the curve f(C) on the right changes as the radius of C increases. In particular, observe how f(C) goes around the origin, and verify that the predictions made by the argument principle are correct.