Solution 7.

Answer.   .  

Solution.   Method I.   Use the implicit formula   .  

Substitute the values given above and get   

                    ,  

                    ,

then simplify and get  

                    .  

Solving for w we obtain

                       

                                          

Therefore,  .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Solution.   Method II.   The general form of a bilinear transformation is

                    ,    and it is not the case that both .

So the desired formula must have one of the following two forms:

either        or    .  

Let us assume that the first form    is the one that works out.

Then we can set up three equations to solve    for  :   

                    ,   
       
then simplify these equations get the system of equations  

                      

Add row 1 to row 3 and get  

                    

Divide row 2 by 1 and subtract it from row 1 to get  

                    

Use      to rewrite the second equation as      and then obtain   .

Use      to rewrite the third equation as      and then obtain   .

Substituting these into      produces the desired result:

                    .

We are really really done.   

Aside.  We can let Mathematica double check our work.

We are really really really done.   

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

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

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

                              The image of the right half-plane    under    is the disk  .

This solution is complements of the authors.

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