Example 10.10.  Show that the transformation    is a one-to-one conformal mapping of the portion of the unit disk    that lies in the upper half-plane    onto the upper half-plane  .  Furthermore, the upper semicircular portion of the boundary is mapped onto the line negative u-axis, and the segment    is mapped onto the positive u-axis.

Figure 10.12  The composite transformation  .

Explore Solution 10.10.

Enter the function  .  

To show    is one-to-one conformal we need to find the inverse function.  Since there is two branches, one of them is appropriate for this problem.

The image is traced using a graph.

We see that the transformation    maps the portion of the unit disk    that lies in the upper half-plane    onto the upper half-plane  .

(c) 2006 John H. Mathews, Russell W. Howell