Exercises for Section 6.5.  Integral Representations for Analytic Functions

          Recall that  [Graphics:Images/IntegralRepresentationModHome_gr_1.gif]  denotes the positively oriented circle  [Graphics:Images/IntegralRepresentationModHome_gr_2.gif].

          Instructions.  The exercises in this section emphasize a solution using either
          
          The Cauchy Integral formula   [Graphics:Images/IntegralRepresentationModHome_gr_3.gif],   or

          Cauchy's Integral formula for derivatives   [Graphics:Images/IntegralRepresentationModHome_gr_4.gif].  

Exercise 1.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_5.gif].  
Solution 1.

 

Exercise 2.  Find    [Graphics:Images/IntegralRepresentationModHome_gr_45.gif].  
Solution 2.

 

Exercise 3.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_85.gif].  
Solution 3.

 

Exercise 4.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_128.gif].  
Solution 4.

 

Exercise 5.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_168.gif].  
Solution 5.

 

Exercise 6.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_211.gif].  
Solution 6.

 

Exercise 7.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_251.gif].  
Solution 7.

 

Exercise 8.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_294.gif]  along the following contours C:  

8 (a).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_295.gif].
Solution 8 (a).

 

8 (b).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_304.gif].
Solution 8 (b).

 

Exercise 9.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_347.gif],  where n is a positive integer.  
Solution 9.

 

Exercise 10.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_366.gif]  along the following along the following contours C:  

10 (a).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_367.gif].
Solution 10 (a).

 

10 (b).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_411.gif].
Solution 10 (b).

 

Exercise 11.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_452.gif][Graphics:Images/IntegralRepresentationModHome_gr_453.gif].  
Solution 11.

 

Exercise 12.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_515.gif]  along the following contours C:  

12 (a).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_516.gif].  
Solution 12 (a).

 

12 (b).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_556.gif].  
Solution 12 (b).

 

Exercise 13.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_631.gif]  along the following contours C:  

13 (a).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_632.gif].  
Solution 13 (a).

 

13 (b).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_673.gif].  
Solution 13 (b).

 

Exercise 14.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_713.gif].  
Solution 14.

 

Exercise 15.  Find  [Graphics:Images/IntegralRepresentationModHome_gr_756.gif]  along the following contours C:  

15 (a).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_757.gif].  
Solution 15 (a).

 

15 (b).  The contour  C is the circle  [Graphics:Images/IntegralRepresentationModHome_gr_797.gif].  
Solution 15 (b).

 

Exercise 16.  Let  [Graphics:Images/IntegralRepresentationModHome_gr_837.gif].  

Find  [Graphics:Images/IntegralRepresentationModHome_gr_838.gif],  where n is a positive integer.  
Solution 16.

 

Exercise 17.  Let  [Graphics:Images/IntegralRepresentationModHome_gr_865.gif]  be two complex numbers that lie interior to the simple closed contour C with positive orientation.  

Evaluate  [Graphics:Images/IntegralRepresentationModHome_gr_866.gif].  
Solution 17.

 

Exercise 18.  Let f be analytic in the simply connected domain D and let  [Graphics:Images/IntegralRepresentationModHome_gr_909.gif]  be two complex numbers

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

Show that

                    [Graphics:Images/IntegralRepresentationModHome_gr_910.gif]

State what happens when [Graphics:Images/IntegralRepresentationModHome_gr_911.gif].  
Solution 18.

 

Exercise 19.  The Legendre polynomial  [Graphics:Images/IntegralRepresentationModHome_gr_956.gif]  is defined by  

                    [Graphics:Images/IntegralRepresentationModHome_gr_957.gif].  

Use Cauchy's integral formula to show that  

                    [Graphics:Images/IntegralRepresentationModHome_gr_958.gif]

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  [Graphics:Images/IntegralRepresentationModHome_gr_967.gif]  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