Exercise 1.  Let    and    be the two branches of the square root function given by Equations (2-28) and (2-29), respectively.  

(2-28)            ,  

(2-29)            ,    

            where    and   so that  .  

Use the polar coordinate formulas in Section 2.2 to find the image of

1 (a).  Quadrant II,  ,  under the mapping  .  

Solution 1 (a).

See text and/or instructor's solution manual.

Answer.  The sector .

Solution.  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 Quadrant II ;    is  

  which can be written as  .  

We are done.   

Aside.  We can let Mathematica double check our work.

                    The mapping  .

                    The domain set    and range set  .

This solution is complements of the authors.

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