Example 10.3. Show that maps
the unit disk |z| < 1 one-to-one and onto the upper half plane Im w > 0.

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 thoughts.

The Mobius transformation .

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(c) John Mathews, 1998, 2006