Exercises for Section 8.1.  The Residue Theorem

Exercise 1.  Find   [Graphics:Images/ResidueCalcModHome_gr_1.gif]   for the following functions:

1 (a).   [Graphics:../Images/ResidueCalcModHome_gr_2.gif].
Solution 1 (a).

 

1 (b).   [Graphics:Images/ResidueCalcModHome_gr_31.gif].
Solution 1 (b).

 

1 (c).   [Graphics:Images/ResidueCalcModHome_gr_65.gif].
Solution 1 (c).

 

1 (d).   [Graphics:Images/ResidueCalcModHome_gr_112.gif].
Solution 1 (d).

 

1 (e).   [Graphics:Images/ResidueCalcModHome_gr_149.gif].
Solution 1 (e).

 

1 (f).   [Graphics:Images/ResidueCalcModHome_gr_188.gif].
Solution 1 (f).

 

1 (g).   [Graphics:Images/ResidueCalcModHome_gr_222.gif].
Solution 1 (g).

 

1 (h).   [Graphics:Images/ResidueCalcModHome_gr_238.gif].
Solution 1 (h).

 

1 (i).   [Graphics:Images/ResidueCalcModHome_gr_263.gif].
Solution 1 (i).

 

1 (j).   [Graphics:Images/ResidueCalcModHome_gr_279.gif].
Solution 1 (j).

 

1 (k).   [Graphics:Images/ResidueCalcModHome_gr_295.gif].
Solution 1 (k).

 

1 (l).   [Graphics:Images/ResidueCalcModHome_gr_333.gif].
Solution 1 (l).

 

1 (m).   [Graphics:Images/ResidueCalcModHome_gr_373.gif].
Solution 1 (m).

 

1 (n).   [Graphics:Images/ResidueCalcModHome_gr_402.gif].
Solution 1 (n).

 

Exercise 2.  Let  [Graphics:Images/ResidueCalcModHome_gr_434.gif]  have an isolated singularity at  [Graphics:Images/ResidueCalcModHome_gr_435.gif].  

Show that   [Graphics:Images/ResidueCalcModHome_gr_436.gif].  

Solution 2.

 

Exercise 3.  Evaluate  

3 (a).  [Graphics:Images/ResidueCalcModHome_gr_443.gif].
Solution 3 (a).

 

3 (b).  [Graphics:Images/ResidueCalcModHome_gr_477.gif].
Solution 3 (b).

 

3 (c).  [Graphics:Images/ResidueCalcModHome_gr_518.gif].
Solution 3 (c).

 

3 (d).  [Graphics:Images/ResidueCalcModHome_gr_568.gif].
Solution 3 (d).

 

3 (e).  [Graphics:Images/ResidueCalcModHome_gr_605.gif].
Solution 3 (e).

 

3 (f).  [Graphics:Images/ResidueCalcModHome_gr_644.gif].
Solution 3 (f).

 

3 (g).  [Graphics:Images/ResidueCalcModHome_gr_679.gif].
Solution 3 (g).

 

Exercise 4.  Let  [Graphics:Images/ResidueCalcModHome_gr_713.gif]  be analytic at [Graphics:Images/ResidueCalcModHome_gr_714.gif].  

If  [Graphics:Images/ResidueCalcModHome_gr_715.gif]  and  [Graphics:Images/ResidueCalcModHome_gr_716.gif]  has a simple zero at  [Graphics:Images/ResidueCalcModHome_gr_717.gif],  then show that  [Graphics:Images/ResidueCalcModHome_gr_718.gif].  

Solution 4.

 

Exercise 5.  Find  [Graphics:Images/ResidueCalcModHome_gr_725.gif],   when  

5 (a).  [Graphics:Images/ResidueCalcModHome_gr_726.gif].  
Solution 5 (a).

 

5 (b).  [Graphics:Images/ResidueCalcModHome_gr_761.gif].  
Solution 5 (b).

 

Exercise 6.  Find  [Graphics:Images/ResidueCalcModHome_gr_811.gif],   when  

6 (a).  [Graphics:Images/ResidueCalcModHome_gr_812.gif].  
Solution 6 (a).

 

6 (b).  [Graphics:Images/ResidueCalcModHome_gr_858.gif].      
Solution 6 (b).

 

Exercise 7.  Find  [Graphics:Images/ResidueCalcModHome_gr_921.gif],   when  

7 (a).  [Graphics:Images/ResidueCalcModHome_gr_922.gif].  
Solution 7 (a).

 

7 (b).  [Graphics:Images/ResidueCalcModHome_gr_958.gif]
Solution 7 (b).

 

Exercise 8.  Find  [Graphics:Images/ResidueCalcModHome_gr_992.gif],   when  

8 (a).  [Graphics:Images/ResidueCalcModHome_gr_993.gif].  
Solution 8 (a).

 

8 (b).  [Graphics:Images/ResidueCalcModHome_gr_1027.gif].  
Solution 8 (b).

 

Exercise 9.  Use residues to find the partial fraction representations of  

9 (a).  [Graphics:Images/ResidueCalcModHome_gr_1072.gif].
Solution 9 (a).

 

9 (b).  [Graphics:Images/ResidueCalcModHome_gr_1088.gif].
Solution 9 (b).

 

9 (c).  [Graphics:Images/ResidueCalcModHome_gr_1104.gif].
Solution 9 (c).

 

9 (d).  [Graphics:Images/ResidueCalcModHome_gr_1122.gif].
Solution 9 (d).

 

9 (e).  [Graphics:Images/ResidueCalcModHome_gr_1151.gif].
Solution 9 (e).

 

9 (f).  [Graphics:Images/ResidueCalcModHome_gr_1169.gif].
Solution 9 (f).

 

Exercise 10.  Let  [Graphics:Images/ResidueCalcModHome_gr_1199.gif]  be analytic in a simply connected domain  D,  and let  C  be a simple closed positively oriented contour in  D.  

If  [Graphics:Images/ResidueCalcModHome_gr_1200.gif]  is the only zero of  [Graphics:Images/ResidueCalcModHome_gr_1201.gif]  in  D  and  [Graphics:Images/ResidueCalcModHome_gr_1202.gif]  lies interior to  C,  then show that

                    [Graphics:Images/ResidueCalcModHome_gr_1203.gif],   where  k  is the order of the zero at  [Graphics:Images/ResidueCalcModHome_gr_1204.gif].  
Solution 10.

 

Exercise 11.  Let [Graphics:Images/ResidueCalcModHome_gr_1213.gif]  be analytic at the points  [Graphics:Images/ResidueCalcModHome_gr_1214.gif].  

If   [Graphics:Images/ResidueCalcModHome_gr_1215.gif],   then show that   [Graphics:Images/ResidueCalcModHome_gr_1216.gif]   for  [Graphics:Images/ResidueCalcModHome_gr_1217.gif].  

Hint:  The solution is a little tricky.  You might need to use  [Graphics:Images/ResidueCalcModHome_gr_1218.gif].  

Solution 11.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2012 John H. Mathews, Russell W. Howell