Exercise 1.  Find the image of the horizontal line   under the reciprocal transformation  .  

Make sketches and indicate the points    in the z-plane and their images    in the w-plane.

Solution 1.

See text and/or instructor's solution manual.

Answer.  The circle  .  

The points   are mapped onto  .

Graph of the horizontal line  ,  using the parametric equations  ;  and the image circle    using the parametric
equations  ,   and   .  

Solution using algebra.

      The inverse mapping is  .   

Now use the substitution    and get:

This is an equation of the circle  .  

We are done.   

                    The horizontal line   and the image circle  ,  divide the z-plane and w-plane into two pieces.

Solution using point images.  

      The points   lie on the horizontal line   in the z-plane.  

Their image points are  ,  and these three points    determine the circle

   in the w-plane.  

If we observe that the center of the circle    is  ,  then an easy calculation shows that

,  

,   and  

,  

and we have verified this observation .  

Aside.  We can let Mathematica double check our work.

                    The horizontal line   and the image circle  .

                    The points   and  .

This solution is complements of the authors.

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