Example 10.11.  Consider the function  ,  which is the composition of the functions    and    where the branch of the square root is  ,  where   ,  ,  and  .  Then the transformation    maps the upper half-plane    one-to-one and onto the upper-half plane    slit along the segment  .  

Figure 10.13  The composite transformation    and the intermediate steps    and  .

Explore Solution 10.11.

Enter the function  .  

To show  w = f(z) =   is one-to-one conformal we need to find the inverse function.

However, in Mathematica  the branch cut for square root is along the negative x-axis.  For graphing purposes, we need the version of the function f(z) with a branch cut along the positive x-axis  .  

The image is traced using a graph.

We see that the transformation    maps the upper half-plane    one-to-one and onto the upper-half plane    slit along the segment  .  

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