Example 8.2.  Find  [Graphics:Images/ResidueCalcMod_gr_23.gif]  if  [Graphics:Images/ResidueCalcMod_gr_24.gif].  

Explore Solution 8.2.

[Graphics:../Images/ResidueCalcMod_gr_29.gif]




[Graphics:../Images/ResidueCalcMod_gr_30.gif]

 

 

Enter the function  [Graphics:../Images/ResidueCalcMod_gr_31.gif]  and construct the Laurent series about  z = 0.

[Graphics:../Images/ResidueCalcMod_gr_32.gif]


[Graphics:../Images/ResidueCalcMod_gr_33.gif]

 

 

The coefficient of  [Graphics:../Images/ResidueCalcMod_gr_34.gif]  so the residue is  [Graphics:../Images/ResidueCalcMod_gr_35.gif].  

[Graphics:../Images/ResidueCalcMod_gr_36.gif]

[Graphics:../Images/ResidueCalcMod_gr_37.gif]

 

 

Aside. We can verify this computation with Mathematica's built in "Residue" procedure.

[Graphics:../Images/ResidueCalcMod_gr_38.gif]


[Graphics:../Images/ResidueCalcMod_gr_39.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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