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

1 (a).   .

Solution 1 (a).

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 pole at    lies inside  .

                              The point    lies inside  .

      We can calculate the winding number using the residue calculus

                       

Here the denominator of  f(z) has a factor of the form  ,  and by Theorem 8.2  .  

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

                        

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.   

Aside.  We can use    and parameterize the integral     using  .  

We are really really really done.   

Aside.  Just for fun, we can investigate the image of the circle  .  

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

                    winds around the origin in the w-plane    times.

Question.   Can you explain what happens to the image curve  f(C)  of the circle  ?

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

                    passes through the origin in the w-plane 4 times.

This solution is complements of the authors.

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