Remark 8.1.  If a repeated root occurs, then the process is similar, and it is easy to show that if P(z) has degree of at most 2, then

            [Graphics:Images/ResidueCalcMod_gr_202.gif],  
where
            [Graphics:Images/ResidueCalcMod_gr_203.gif]   

 

Example 8.9.  Express  [Graphics:Images/ResidueCalcMod_gr_204.gif]  in partial fractions.  

Explore Solution 8.9.

Enter the function  [Graphics:../Images/ResidueCalcMod_gr_212.gif] and find the partial fraction expansion.  

[Graphics:../Images/ResidueCalcMod_gr_213.gif]





[Graphics:../Images/ResidueCalcMod_gr_214.gif]

 

 

 

Compare this with Mathematica's Apart  procedure for the partial fraction expansion.

[Graphics:../Images/ResidueCalcMod_gr_215.gif]



[Graphics:../Images/ResidueCalcMod_gr_216.gif]

 

 

In the next example we will see that our subroutine will work for complex factors and "Apart" does not.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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