Exercise 1.  State where the following mappings are conformal.  

1. (e)   .  

Solution 1 (e).

See text and/or instructor's solution manual.

Answer.     is conformal for all except  .  

Solution.   Calculate and determine where  .  

                       

It is easy to see that  .  

However,    is not defined at the point  .  

Therefore,   is conformal for all 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  .  

Remarks.  In Section 2.5 we saw that the reciprocal transformation    maps "generalized circles" onto "generalized circles."

In Section 10.2 we will see that the transformation    will also map "generalized circles" onto "generalized circles."

This solution is complements of the authors.

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