Solution 5.

Answer.   The image of the upper half plane    is the region  .

Solution.  Method I.   The boundary of upper half plane    is the real axis  ,  

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

The image points  ,  

give the unit circle    a positive orientation and the region    a left orientation.

Therefore, the image of the upper half plane    under    is  .  

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

then    lies in the region   ,

which leads us to conclude that the image of the upper half plane    is the region  .

We are done.   

Solution.  Method II.   Start by finding the inverse transformation for   .

Use equations (10-13)  and  (10-14).  

(10-13)             ,

(10-14)             .

Here we have    and   .  

Then  

                         
            
Hence, the inverse transformation is   .  

Then get  

                       

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

Therefore, the image of the upper half plane    under    is  .  

We are really done.   

We are really really done.   

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

                              The image of the upper half plane    under    is the region  .

This solution is complements of the authors.

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