Exercise 17.  Show that the transformation    maps the disk    onto the right half-plane  .  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images   in the w-plane.

Solution 17.

See text and/or instructor's solution manual.

Solution using algebra.

      The map     has the inverse  .   Substitute    into    and get:

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

We are done.   

                    The disk    and the image right half-plane  .  

                    The points    and  .

Solution using point images.  

      The points    lie on the circle   in the z-plane.  

Observe that their image points are  ,  and these three points    

determine the vertical line    in the w-plane.  

An easy calculation shows that  ,  ,    and we have verified this observation.  

This verifies our claim that the image of the circle    is the vertical line    in the w-plane.  

Furthermore, we observe that the point    is mapped onto the point     so the image region lies to the right of    in the w-plane.  

Therefore, the image of the disk    is the right half-plane  .  

Aside.  We can let Mathematica double check our work.

                    The disk    and the image right half-plane  .  

                    The points    and  .

This solution is complements of the authors.

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