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


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  


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  [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