Example 8.1.  If  [Graphics:Images/ResidueCalcMod_gr_11.gif],  then the Laurent series of f about the point [Graphics:Images/ResidueCalcMod_gr_12.gif] has the form   

            [Graphics:Images/ResidueCalcMod_gr_13.gif],  and

            [Graphics:Images/ResidueCalcMod_gr_14.gif].  

Explore Solution 8.1.

Enter the function  [Graphics:../Images/ResidueCalcMod_gr_15.gif]  and construct the Laurent series about z = 0.

[Graphics:../Images/ResidueCalcMod_gr_16.gif]



[Graphics:../Images/ResidueCalcMod_gr_17.gif]

 

 

 

The coefficient of  [Graphics:../Images/ResidueCalcMod_gr_18.gif]  is  2  so the residue is Res[f, 0] = 2.

[Graphics:../Images/ResidueCalcMod_gr_19.gif]

[Graphics:../Images/ResidueCalcMod_gr_20.gif]

 

 

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

[Graphics:../Images/ResidueCalcMod_gr_21.gif]


[Graphics:../Images/ResidueCalcMod_gr_22.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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