The alternate definition of  .  The alternate definitions   and    for two branches of the square root are given in Exercises 2 and 3.

Exercise 3.  Let    in Equation (2-30).   Find the range of the function   .  

Solution 3.

See text and/or instructor's solution manual.

Answer.  Since  ,  where  ,   and     (explain!),

we see that the point     will lie in the lower half plane plus the positive real axis.

Thus, the range of   is  .  

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

Then  ,  where    and  ,  

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 is    .  

We are done.   

Aside.  We can let Mathematica double check our work.

                    The mapping  ,  where    and  .  

                    The domain set    and range set  .

This solution is complements of the authors.

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