Exercise 1.  State where the following mappings are conformal.  

1. (d)   .  

Solution 1 (d).

See text and/or instructor's solution manual.

Answer.     is conformal for all except  .  

Solution.   Calculate and determine where  .  

                       

Since     for all  ,     only 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  .  

We are really really done.

Caveat.  At a point    where     we will have

                    .

Applying Theorem 10.2 we see that the mapping   magnifies angles at the vertex    by the factor  .

                      For     at    we have      and   ,  and
                      the mapping   magnifies angles at the vertex    by the factor  .

Remark.  In Section 10.3 and Section 10.4 we will study some composite mappings involving  .

This solution is complements of the authors.

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