In this video, we show mappings from s to L(s) when L(s) has a single pole.

In this first example L(s)=10/s, so L(s) has a pole at the origin, shown on the left.

As with the mapping of zeros, here we traverse a circular path in "s" in the clockwise direction.  The angle in between the pole of L(s) and the path "s" is shown in blue.

In this case, however, The path in L(s) moves in the counterclockwise direction and is shown in blue.  The reason for this shift in direction from the previous examples is described in the text and the equations accompanying the first example that was done using a single pole at the origin.  The important thing to note here is that while the path in "s" encircles the pole one time in the clockwise direction, the path in L(s) encircles the origin one time in the opposite, or counterclockwise, direction.

If we move the pole to -4 so that L(s)=1/s+4, we can see that the path in L(s) changes.  What is important is that since the path in s encircles the pole in the clockwise direction, the path in L(s) still encircle the origin in the counterclockwise direction.

If we move the pole even further from the origin so L(s)=1/s+6, we see that the path in s does not encircle the pole, so the path in L(s) does not encircle the origin.

This last case looks a little more complicated.  In this case the path in s comes very close to the pole in L(s), and the magnitude L(s) becomes very large.  In these graphs any value with a magnitude greater than 12 is truncated, as is shown here.  However, as before, since s encircles the zero in the clockwise direction, L(s) encircles the origin in the counterclockwise direction.