Example 10.7. Find the bilinear transformation w = S(z) that maps
the crescent-shaped region that lies inside the disk |z-2| < 2 and outside the circle
|z-1| = 1 onto a horizontal strip.

Solution. For convenience we choose
which are mapped onto the points , respectively.
In this case we remove the terms involving , in the implicit formula,
because this implies that .

Check our work and look at the images of .

And we need to look at the points .

To illustrate the mapping, consider the inverse image of the horizontal strip,
and the mapping .

The Mobius transformation w = (-iz + 4i)/z. 

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(c) John Mathews, 1998, 2006