Argument Principle

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The image below shows how the function h(z) = z^(-2)-z^(-4)$ maps the positively-oriented circle |z|=2 so that it winds around the origin twice clockwise. Note that h(z) has a pole of order four at z=0 and two zeroes of order one at z = 1 and -1. So this is exactly what the argument principle predicts should happen.

The Sage source code that generated the image is given below:

var('t'); z = 2*exp(I*t); #this is the parametrization of the circle. h = z^(-2)-z^(-4); A = animate([parametric_plot((h.real(),h.imag()),k,k+0.5,color='red') for k in xsrange(0,2*pi-0.3,0.3)], xmin=-1, xmax=1, ymin=-1, ymax=1); A.show(iterations=1); #A.gif(iterations=1); #this will create a .gif file with the animation.