In this video, we show mappings from s to L(s) when L(s) has multiple poles and zeros.  The path s is still a circle of radius 5, as before.

In this first example L(s) has two poles, at s=-1, and s=-3, and a zero at s=-2.  All of these are encircled by the path in s.  As the path in s traverses a circle the angles to the zeros in L(s) are shown in pink, the angles to the poles are shown in blue.  The angle between the origin and L(s), on the right side, is simply the sum of the angles to the zeros (the pink arcs on the left) minus the angles to the poles (the blue arcs on the left).  Since this result is always negative in this case as s moves in a clockwise direction, the angle between the origin and L(s) will always be in the counterclockwise direction, and is shown in blue on the right.

This next example has one of the poles outside of the path of s.  Since s, on the left graph, now encircles one pole and one zero of L(s), the net angle of L(s), on the right graph is zero.

This example shows both poles outside of the path of s.  Since s, on the left, encircles a single zero in the clockwise direction, L(s), on the right, encircles the origin once in the clockwise direction.

The next example shows two poles within the path of s, and no zeros.  Since the path of s, on the left, encircles two zeros in the clockwise direction, the path of L(s), on the right, encircles the origin twice in the counterclockwise direction.

So far all of our examples have had poles and zeros on the real axis, but there is no requirement that this be true, as shown in this example with two complex conjugate poles that are encircled.

This last example shows two complex conjugate poles that are not encircled.  Note that the path in L(s) is very complex, but that it does not encircle the origin.