Example 10.9. The transformation is a one-to-one
conformal mapping of the unit disk onto the horizontal strip |v| < .
Furthermore, the upper semicircle of the disk is mapped onto the line
and the lower semicircle onto .

To show is one-to-one conformal we need to find the inverse function.

The image is traced using a graph.

 Thus, the image of the unit disk |z| < 1 is the horizontal strip |v| < .

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(c) John Mathews, 1998, 2006