Solution 11.

Answer.  The image of the lower half plane    under    is the unit disk  .   

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

and we can give the lower 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 the lower half plane    under    is  .  

Furthermore, as a double-check we can choose the point    in the lower 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.   

Aside.  We can let Mathematica double check our work.

The boundary of the lower half plane is the real axis, 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 unit disk a positive orientation.
So the image of the lower half plane is the unit disk .

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

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      and we get  

                      

                      

                      

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

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 lower half plane    under    is the unit disk  .   

This solution is complements of the authors.

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