Exercise 9.  Use the results of  Exercise 8  with    to evaluate  

9 (a).  .

Solution 9 (a).

See text and/or instructor's solution manual.

Answer.  .

Solution.  Parametrize    with      for   .  

Then   ,   and      and we obtain  

                      

                    The contour      for   .  

We are really done.  

Aside.  After we have developed the Cauchy-Goursat Theorem in Section 6.3,

we will introduce Corollary 6.1, .  

If you have proved this exercise you are ready for more advanced methods to compute contour integrals!

This solution is complements of the authors.

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