Solution 6.

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 last equation is easy to solve and we get      and then the first equation yields   .

Use these values to rewrite the second 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 right half-plane    under    is the upper half-plane  .

                                   The image of the disk    under    is the region  .

This solution is complements of the authors.

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