Exercises for Section 6.5.  Integral Representations for Analytic Functions

          Recall that    denotes the positively oriented circle  .

          Instructions.  The exercises in this section emphasize a solution using either
          
          The Cauchy Integral formula   ,   or

          Cauchy's Integral formula for derivatives   .  

Exercise 1.  Find  .  
Solution 1.

Exercise 2.  Find    .  
Solution 2.

Exercise 3.  Find  .  
Solution 3.

Exercise 4.  Find  .  
Solution 4.

Exercise 5.  Find  .  
Solution 5.

Exercise 6.  Find  .  
Solution 6.

Exercise 7.  Find  .  
Solution 7.

Exercise 8.  Find    along the following contours C:  

8 (a).  The contour  C is the circle  .
Solution 8 (a).

8 (b).  The contour  C is the circle  .
Solution 8 (b).

Exercise 9.  Find  ,  where n is a positive integer.  
Solution 9.

Exercise 10.  Find    along the following along the following contours C:  

10 (a).  The contour  C is the circle  .
Solution 10 (a).

10 (b).  The contour  C is the circle  .
Solution 10 (b).

Exercise 11.  Find  .  
Solution 11.

Exercise 12.  Find    along the following contours C:  

12 (a).  The contour  C is the circle  .  
Solution 12 (a).

12 (b).  The contour  C is the circle  .  
Solution 12 (b).

Exercise 13.  Find    along the following contours C:  

13 (a).  The contour  C is the circle  .  
Solution 13 (a).

13 (b).  The contour  C is the circle  .  
Solution 13 (b).

Exercise 14.  Find  .  
Solution 14.

Exercise 15.  Find    along the following contours C:  

15 (a).  The contour  C is the circle  .  
Solution 15 (a).

15 (b).  The contour  C is the circle  .  
Solution 15 (b).

Exercise 16.  Let  .  

Find  ,  where n is a positive integer.  
Solution 16.

Exercise 17.  Let    be two complex numbers that lie interior to the simple closed contour C with positive orientation.  

Evaluate  .  
Solution 17.

Exercise 18.  Let f be analytic in the simply connected domain D and let    be two complex numbers

that lie interior to the simple closed contour C having positive orientation that lies in D.  

Show that

                    

State what happens when .  
Solution 18.

Exercise 19.  The Legendre polynomial    is defined by  

                    .  

Use Cauchy's integral formula to show that  

                    

where C is a simple closed contour having positive orientation and z lies inside C.  
Solution 19.

Exercise 20.  Discuss the importance of being able to define an analytic function    with the contour integral in formula (6-44) the Cauchy Integral Formula.  

How does this definition differ from other definitions of a function that you have learned?  
Solution 20.

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