Exercise 20.  Show that    is mapped onto the point    on the Riemann sphere.  

Solution 20.

See text and/or instructor's solution manual.

Solution.  The cartesian form of the point    in three dimensional space is  .  
The Riemann sphere sphere is centered at    and has radius  .
If we use the notation    for an arbitrary point in three dimensional space then the equation of the Riemann sphere is  .  

      The line segment L from the north pole to the point   .  is given by the vector function

            

and we must find the point     where L intersects the Riemann sphere.  The coordinate functions for   are  

            .

      Substitute these values into   and get:

               

    Now substitute into    and get  .

Therefore,   .

We are done.   

Aside.  The lower half of the Riemann sphere corresponds to the unit disk     and the upper half of the Riemann sphere corresponds to the exterior of the unit circle , i. e. the region  .   

                    The Riemann hemispheres corresponding to the unit disk    and the region  .  

Aside.  Consider the inverse transformation  ,  and the equations

  and  .  When these values are substituted into   we get  .

This solution is complements of the authors.

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