Solution 4.

Solution.  Method I.   In Example 10.3 we showed that    maps the disk    onto the upper half plane  .  

So all that is needed is to to determine the image of the upper half plane    under the mapping   .  

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  ,  

lie on the imaginary axis  ,  and give the right half plane    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 right half plane   ,

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

        Now intersect the images of the disk    and upper half plane  .  

The image of the portion of the disk    that lies in the upper half plane    under    

is the intersection of the upper half-plane    and the right half plane  ,  which is the first quadrant  .

We are done.   

Solution.  Method II.   In Example 10.3 we showed that    maps the disk    onto the upper half plane  .  

So all that is needed is to to determine the image of the upper half plane    under the mapping   .  

The inverse transformation is    and we get  

                       

Then    implies that    which in turn implies that  .  

Hence the image of the upper half plane    under the mapping    is the right half plane  .  

        Now intersect the images of the disk    and upper half plane  .   Therefore,

the image of the portion of the disk    that lies in the upper half plane    under    

is the intersection of the upper half-plane    and the right half plane  ,  which is the first quadrant  .

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    under    is  .

This solution is complements of the authors.

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