Chapter 10. Show that the conformal mapping maps the domain D that is the portion of the right half-plane Re(z)>0 that lies exterior to the circle |z-5|=4 onto the annulus 1 <|w|<2.

To show S(z) is one-to-one, find the inverse function.

To show S(z) is onto we will consider a graph.
The graph is not conclusive, but it appears to confirm our suspicion.

The Mobius transformation . 

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(c) John Mathews, 1998, 2006