Example 8.6.  Find  [Graphics:Images/ResidueCalcMod_gr_131.gif]  where [Graphics:Images/ResidueCalcMod_gr_132.gif] denotes the circle [Graphics:Images/ResidueCalcMod_gr_133.gif] with positive orientation.  

[Graphics:Images/ResidueCalcMod_gr_134.gif]

Explore Solution 8.6.

Enter the function  [Graphics:../Images/ResidueCalcMod_gr_146.gif]  and locate the singularities.  

[Graphics:../Images/ResidueCalcMod_gr_147.gif]




[Graphics:../Images/ResidueCalcMod_gr_148.gif]

 

 

 

Find out which singularities lie within a circle  [Graphics:../Images/ResidueCalcMod_gr_149.gif].  

[Graphics:../Images/ResidueCalcMod_gr_150.gif]




[Graphics:../Images/ResidueCalcMod_gr_151.gif]

[Graphics:../Images/ResidueCalcMod_gr_152.gif]

 

 

 

Compute the residues at  [Graphics:../Images/ResidueCalcMod_gr_153.gif].  

[Graphics:../Images/ResidueCalcMod_gr_154.gif]



[Graphics:../Images/ResidueCalcMod_gr_155.gif]

 

 

The integral is computed by the residue calculus.

[Graphics:../Images/ResidueCalcMod_gr_156.gif]





[Graphics:../Images/ResidueCalcMod_gr_157.gif]

[Graphics:../Images/ResidueCalcMod_gr_158.gif]

[Graphics:../Images/ResidueCalcMod_gr_159.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell