Solution 8.

Answer.   .  

Solution.   Method I.   Use the implicit formula  .  

Substitute the values given above and get   ,  

then simplify and 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  :   

                    ,   

In the third equation we will take reciprocals and write it as   ,  then we have

                    ,   
       
then simplify these equations get

                    .  

Use    to rewrite the second equation as    then solve the system of two equations

                    

Subtracting the first equation from the second equation and get   .

Use      in the first equation and get   .  

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 upper half-plane    under    is the upper half-plane  .

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

This solution is complements of the authors.

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