Exercise 1.  State where the following mappings are conformal.  

1. (b)   .  

Solution 1 (b).

See text and/or instructor's solution manual.

Answer.     is conformal for all except    where    is an integer.  

Solution.   Calculate and determine where  .  

                       

It is known that  ,  where is an integer.  

Therefore,   is conformal for all except    where    is an integer.  

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 10.4 we will study some trigonometric mappings, including  .

This solution is complements of the authors.

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