Exercise 7.  Recall    is the circle of radius  r  centered at  a,  oriented counter-clockwise.  

7 (a).  Evaluate  .  

Solution 7 (a).

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

                       

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.

                    The contour      for   .

The function is    and the curve is    for    and we obtain

                      and  ,  
                    
                    then

                       

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

                      

We are Really done.  

Aside.  After we have developed the Cauchy-Goursat Theorem in Section 6.3 the result will follow immediately:

                    .  

This solution is complements of the authors.

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