Exercise 1.  Let  .  Find the number of times the image    winds around the origin if  

1 (c).   .

Solution 1 (c).

See text and/or instructor's solution manual.

Answer.   .  

Solution.  For    and  ,  the number of times the image    winds around the origin,  

,  can be calculated using Theorem 8.9 (Winding Number)    

                    .  

The denominator of the integrand      can be factored as

                      

Thus,    has simple poles at  ,  ,  ,  ,  and  .

The poles at  ,  ,  ,  ,  and    all lie inside  .

                              The points   lie inside  .

      We can calculate the winding number using the residue calculus

                      

      It will save a lot of effort to establish the following general principle:   

                    If  f(z) has a simple zero at   then  .

This can be proven with the following computation

                                

Now apply this at the points   ,  ,  ,  ,  and    that lie inside    and get  

                      

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

                    The image curve  f(C)  of the circle    under the mapping    

                    winds around the origin in the w-plane    times.

We are really really done.   

      If you do not use this general principle, then the residues are tedious to calculate.  The details are given below.

Here the denominator of    has a factor of the form  ,  and by Theorem 8.2  .  

In this exercise, the limit can be calculated as follows:

                        

Similarly, we calculate the residue at :

                        

Similarly, we calculate the residue at :

                        

Similarly, we calculate the residue at :

                        

Similarly, we calculate the residue at :

                        

We are really really done.   

Aside.  If you want to get a better idea of "winding" then we can investigate the image curve  f(C)  of the circle  .   

This is just for fun !

                    The image curve  f(C)  of the circle    under the mapping    

                    winds around the origin in the w-plane    times.

This solution is complements of the authors.

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