Example 10.6. Show that the mapping
maps the disk |z+1| < 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, use the method of oriented points on the boundary curve.

To find the image of the disk |z+1| < 1 under w = S(z)
we use the change of variable and find the image of |z| < 1 under w = g(z) = S(z-1).

Now plot the graphs.

The Mobius transformation . 

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(c) John Mathews, 1998, 2006