Solution 3.

Answer.  The image of    under    is the unit disk  .  

Solution.  Method I.   The boundary of the right half plane    is the imaginary axis  ,  

and we can give the right half plane   a left orientation by using the points  .

The image points  ,  give the unit circle  

  a positive orientation and the disk    a left orientation.

Therefore, the image of right half plane    under    is  .  

Furthermore, as a double-check we can choose the point    in the right half plane  ,  

then    lies in the unit disk  ,

which leads us to conclude that the image region lies inside the unit circle  .

We are done.   

Solution.  Method II.   In Example 10.3 we found the inverse transformation.  

If we write  ,  then we have  ,  ,  , and .  

Using Equation (10-14), we find that the inverse is given by

            
(10-16)             .  

Then get  

                       

Then    implies that    which implies that  ,  which in turn implies that  .  

Therefore, the image of the right half plane    under the mapping    is the unit disk  .

We are really done.   

Aside.  We can let Mathematica double check our work.

We are really really done.   

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

                                        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