Example 10.3.  Show that    maps the unit disk    one-to-one and onto the upper half-plane  .  

            Figure 10.5  The image    under  ,  the points
              are mapped onto the points  ,  respectively.

Explore Solution 10.3.

Enter the function  .

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

Check out the inverse function.

Consider the unit circle C: |z| = 1.  Since  ,  the image of these points will satisfy  .  
Square both sides of the above equation and obtain   which can be simplified.

The circle  C  divides the z-plane into two portions, and its image, v = 0, is the u-axis which divides the w-plane into two portions. The image of a point interior to  C  will determine which half plane is the image of the unit disk.  For example

Therefore, the image of the unit disk is the upper half plane .  

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

We see that the transformation    maps the unit disk one-to-one and onto the upper half plane .  

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