Exercise 15.  Find the image of the sector   ,   under   .  

Solution 15.

Answer.   The image of the sector      under      is   .

Short Solution.   The image of      under      is the upper half-plane   ,  

then the image of      under      is  the unit disk   .

Solution.   The mapping    can be written as a composition  

                    ,    

where   

                    ,    and   

                    .

The mapping   can be expressed as

                    ,  

                    or       and    .  

Then      implies      implies   .   Here we have   .  

Also,      implies      implies   .  

        Hence, the image of the sector   ,  under the mapping    

is the upper half-plane  .    

        The boundary of the upper half-plane      is the real axis   ,  

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

Then      maps        onto    ,  

which is a positive orientation for the unit circle   .

        Hence, the image of the upper half plane      under      is the unit disk   .  

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

then      lies in the unit disk   ,

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

          Therefore,  image of the sector      under      is   .

We are done.   

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                                              

We are really done.   

Aside.   If it is important then we can include the following details.

          Here the values        and        are computed as limits.  

                    ,  

                    .  

          Similarly, the values        and        are computed as limits.  

                    ,  

                    .  

This solution is complements of the authors.

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