.
Figure 2: Mapping via 
. Region mapped is shown in
the next figure.
Figure 3: The region of the z-plane being mapped. Find the images of the
the solid lines in the figure above.
Any sector of opening angle 
centered around
the origin maps into the whole w-plane.
In fact, any wedge at the origin in z is has its opening angle doubled by this
mapping, but wedges elsewhere have there opening angle preserved.
The polar map in Figure 4 depicts the range,
Lines of constant x,y map into
parabolas.
Practice: find the image of each of the solid lines of Figure 3
in Figure 4.
Mapping is conformal everywhere but at the origin and infinity.
Straight line from 
to 
maps into a
``keyhole'' contour around the origin.
Figure 4: Polar map of the same function.