Exercise 7.  Find a branch of the multivalued cube root function that is different from those in Exercises 5 and 6. State the domain and range of the branch you find.  

Solution 7.

See text and/or instructor's solution manual.

Solution.  The likely choice is  ,  

where  ,  ,  ,  and  .  

Use polar coordinates    in the z-plane and    in the  w-plane.  

Then    

maps the point    in the xy-plane onto the point    in the uv-plane,

and we get the equations    and  .  

Using the equations    and  ,  we find that the image of    is  

  which can be written as  ,

and in standard form   .

We are done.   

Aside.  We can let Mathematica double check our work.

                    The mapping  .  

                    The domain set    and range set  .

Alternate Solution.  The function  ,  where  ,   and    does the job.  

Explain why, and find the range of this function, or of a different function that you concoct.

This solution is complements of the authors.

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