Solution 10.

Solution.  Method I.   The boundary of the unit disk    is the unit circle  ,  

and we can give    a positive orientation and    a left orientation by using the points  .

The image points  ,  give the right half-plane a left orientation.  

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

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

which leads us to conclude that the image region lies to the right of the imaginary axis  .   

We are done.   

Aside.  We can let Mathematica double check our work.

The boundary of the unit disk is the unit circle, and we can give the boundary a positive orientation by using the points ,  ,  and  .  

Check our work and by looking at the images of  .  

The image points  ,  ,  and    give the right half plane a positive orientation.

Furthermore, as a double-check we can choose the point    in the unit disk, then lies in the right half-plane.

We are really 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  

                      

                      

                      

                      

                      

                      

Since      we can conclude that      which implies that   .  

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

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 unit disk    under    is the right half plane  .  

                                   The image of    under    is  .

This solution is complements of the authors.

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