7 (c).  Evaluate  .  (The minus sign in means the clockwise orientation.)

Solution 7 (c).

See text and/or instructor's solution manual.

Answer.   .  

                    The contour      for   .

Solution Method I.  The function is    and the curve is    for    and we obtain

                      and  ,  
                    
                    then

                        

We are done.  

Solution Method II.  We could use a method that uses the following complex computations.

                    The contour      for   .

The function is    and the curve is    for    and we obtain

                      and  ,  
                    
                    then

                       

We are really done.  

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

we will be able to use Corollary 6.1, ,  

and we would immediately conclude that

                    .

This solution is complements of the authors.

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