Exercises for Section 6.3.  The Cauchy-Goursat Theorem

Hint for the exercises.  Many of the computations are an applications of Corollary 6.1:
If C is a simple closed contour with positive orientation such that    lies interior to C, then  .

Exercise 1.  Determine the domain of analyticity for the following functions and evaluate  .  

1 (a).  .  
Solution 1 (a).

1 (b).  .
Solution 1 (b).

1 (c).  .
Solution 1 (c).

1 (d).  .  
Solution 1 (d).

Exercise 2.  Show that  ,  where C is the square with vertices    and having positive orientation.  
Solution 2.

Exercise 3.  Show that  .  
Solution 3.

Exercise 4.  Find      for  

4 (a).  circle      having positive orientation.
Solution 4 (a).

4 (b).  circle      having positive orientation.
Solution 4 (b).

Exercise 5.  Find    for the  

5 (a).  circle    having positive orientation.
Solution 5 (a).

5 (b).  circle    having positive orientation.  
Solution 5 (b).

Exercise 6.  Let C be the triangle with vertices    and having positive orientation.  Parametrize C and show that  

6 (a).  .  
Solution 6 (a).

6 (b).  .  
Solution 6 (b).

Exercise 7.  Evaluate       for

7 (a).  the circle    having positive orientation.
Solution 7 (a).

7 (b).  the circle    having positive orientation.
Solution 7 (b).

7 (c).  the circle    having positive orientation.
Solution 7 (c).

Exercise 8.  Use Green's theorem to show that the area enclosed by a simple closed contour C is   .  
Solution 8.

Exercise 9.  Parametrize    with  ,  for  .   Use the
principal branch of the square root function:  ,  for  ,  to find   .  
Hint:  Take limits as  .  
Solution 9.

Exercise 10.  Evaluate       for the contours shown in Figure 6.29 (a) and 6.29 (b).  

                         
                         Figure 6.29 (a)  The contour  .                                   Figure 6.29 (b)  The contour  .

Solution 10 (a).

Solution 10 (b).

Exercise 11.  Evaluate    .  
Solution 11.

Exercise 12.  Suppose that    is analytic for all values of  .  Show that  

                    .  

Hint.  Integrate    around the circle  .   
Solution 12.

Exercise 13.  Let C be the figure eight contour shown in Figure 6.28(a).   
                             
          Figure 6.28  The points    and    which lie inside the contour  .

13 (a).  Evaluate    .  
Solution 13 (a).

13 (b).  Evaluate    .  
Solution 13 (b).

Exercise 14.  Compare the various methods for evaluating contour integrals.  What are the limitations of each method?  
Solution 14.

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