Exercise 9.  Describe a Riemann surface for the domain of definition of the multivalued function  

9 (a).    .  

Solution 9 (a).

See text and/or instructor's solution manual.

Answer.  For    we have  

        ,  

as the three branches of the cube root with domains  .  

As in the text, form three domains which are copies of the complex plane slit along the negative real axis,  and stack    directly above each other.  

                                                            The image of .

                                                            The image of .

                                                            The image of .

               Join the edge of in the upper half plane to the edge of in the lower half plane.  

               Join the edge of in the upper half plane to the edge of in the lower half plane.

It is possible to make above construction, it would look something like a spiral parking ramp.

                    The three sheets    of the Riemann surface domain for the cube root function  .  

Finally, join the edge of in the upper half plane to the edge of in the lower half plane.  

                    The Riemann surface domain for the cube root function  .  

Background for the Solution.  In Exercise 5, Exercise 6, and Exercise 7 we investigated three branches of the cube root function  .  

For each branch we used the coordinate notation in the same domain set  .  

Thus we obtained three branches of  :  

       with
    
    ,   for  .  

The images of  D  under    is the sector  ,   for  .  

Important note.  In Exercise 5, Exercise 6, and Exercise 7 we used the same domain set   and three different formulas   for the functions.

In the construction of the Riemann surface for we will use three different domain sets and one formula    for the function.  

As in the text, each domain     is a copy of the z-plane slit along the negative real axis.

Solution for the Riemann Surface.  

     To construct the Riemann surface for    we need to consider extending Equation (2-30) for the case of the cube root.

For each branch we must use it's own special coordinate notation for    in the z-plane, i.e. use the special special domain set  

           with  
    
        .  

The image of the domain set    is the sector  ,   for  .  

       Notice that in the w-plane the angles for the three sectors are  ,  ,  and  

and taken all together these angles cover the interval  .   

So when we glue together the image sectors    they cover the entire complex w-plane.

      The domain set will be a Riemann surface, and it is formed as follows:

Join the edge of in the upper half plane to the edge of in the lower half plane.  

Join the edge of in the upper half plane to the edge of in the lower half plane.

      It is possible to make above construction, it would look something like a spiral parking ramp.

                    The image of    under    is the sector  .

                    The image of    under    is the sector  .

                    The image of    under    is the sector  .

                    The three sheets    of the Riemann surface domain for the cube root function  .  

           Finally, join the edge of in the upper half plane to the edge of in the lower half plane.  

This final connection makes the a construction in the real world of 3D space difficult.

                    The Riemann surface domain for the cube root function  .  

This solution is complements of the authors.

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