Exercise 11.  Find the image of the disk    under  .  

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 11.

See text and/or instructor's solution manual.

Answer.  The region that lies exterior to the unit circle  ,  i.e.   .

Solution using algebra.

      The inverse mapping is  .   

Now use the substitution    and get:

This is the region that lies exterior to the unit circle  ,  

i.e.  the image region is  .

We are done.   

                    The disk    and the image region  .

                    The points    and  .

Solution using point images.  

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

The image points lie on the circle    in the w-plane,

and this is shown by observing that the center of the circle    is  ,  and then making the computations:  

        ,  

        ,  and  

        .

This verifies our claim that the image of the circle    is the circle  .  

Furthermore, we observe that the point    is mapped onto the point     so the image region lies outside the circle .

Therefore, the image of the disk    is the region  .

Aside.  We can let Mathematica double check our work.

                    The disk    and the image region  .

                    The points    and  .

This solution is complements of the authors.

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