Example 2.23.  For the transformation  ,  find the image of the portion of the right half plane    that lies inside the closed disk  .  

Explore Solution 2.23.

First, find the image of the right half plane  .  

This last inequality    is the same as    and is the disk of radius 1 centered at  w = 1  in the w-plane.

Now find the image of the interior of the circle.  

This last inequality    is the same as    and is the exterior of the circle with center    and radius    in the w-plane.

Use Mathematica to graph the mapping.

We see that the image of the portion of the right half plane    that lies inside the circle  under the reciprocal transformation    is the crescent shaped region in the w-plane which is the portion of the disk    that lies outside the circle  .  

(c) 2006 John H. Mathews, Russell W. Howell