The argument principle

This applet takes a rational function f(z) = (z^2 + az + b) / (z^2 + cz + d), where a, b, c, d are adjustable real numbers, and displays its zeroes (the cyan crosses) and the poles (the black crosses).  It also shows a circle (of radius 2 around i) and its image.  Observe that the number of zeroes inside the circle, minus the number of poles, equals the number of times the image winds anticlockwise around the origin.  This is the argument principle, and it holds true for any simple closed curve, not just the circle.  Try creating some closed curves by dragging the mouse on the left half-plane to check this.

You can use the < and > buttons to manipulate the rational function, or edit the co-efficients directly.  Note what happens when a zero or pole passes through the circle.  Also notice that when two zeroes collide, they tend to bounce off in a perpendicular direction, and similarly for the poles.