Exercise 5.  Evaluate    from    along the following contours, as shown in Figures 6.13 (a)  and  6.13 (b).  

5 (b).  The contour C that is the upper half of the circle  ,   oriented clockwise,   as shown in Figures 6.13 (b).

                                   Figure 6.13 (a).  The contour  .  

Solution 5 (b).

See text and/or instructor's solution manual.

Answer.  .  

                    Figure 6.13 (b).  The contour      for   .

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

                       and   ,   
                    
                    then

                       

The last two real integrals are computed using

                      

                    and

                      

We are done.  

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

                    Figure 6.13 (b).  The contour      for   .

The function is      and the curve is    for    and

from Section 1.2 we apply the formula   .   Then we obtain

                       and   ,  
                    
                    then

                       

Here we have used the calculations indicated by equation (6-8) in Section 6.1:

                       

This solution is complements of the authors.

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