Exercise 1.  State where the following mappings are conformal.  

1. (f)   .  

Solution 1 (f).

See text and/or instructor's solution manual.

Answer.     is conformal for all z except  .  

Solution.   Calculate and determine where  .  

                       

It is easy to see that  .  

However,    is not defined at the point  .  

Therefore,    is conformal for all z except  .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  We can let Mathematica illustrate our work.

                    A small portion of the mapping  .  

                    Another small portion of the mapping  .  

                    Yet another small portion of the mapping  .  

Remark.  In Section 10.2 we will see that a transformation of the form    will map "generalized circles" onto "generalized circles."

This solution is complements of the authors.

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