Solution 2.

Answer.   .  

Solution.   Method I.   Use equations (10-13)  and  (10-14).  

(10-13)             ,

(10-14)             .

Here we have    and   .  

Then  

               

Therefore,   .  

We are done.   

Solution.   Method II.   Use brute force and solve the equation    for  z  and get

                    

Therefore,   .  

We are really done.   

Aside.  We can let Mathematica double check our work.

We are really really done.   

Aside.  We can look at some graphs of the mapping  .

               The image of the unit disk    under    is the right half plane  .  
               You will be asked to show this in Exercise 10.

                    The image of the lower half plane    under    is the unit disk  .  
                    You will be asked to show this in Exercise 11.

                                        The image of the left half-plane    under    is the upper half-plane   .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell