Converting to Partial Fractions
 

Short Cut Methods
solve for the roots of ansn++a1s+a0=0 say: -r1, -r2, , -rthen
    1. For each real (distinct) root, say -ri ,include a term


        where
    1. For each repeated root, e.g. -rq repeated k times, include k terms
      where
    1. for each complex conjugate pair of roots, e.g. -a+ib and a-ib, include the following pair of terms,


where



 
 
 

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Example :
 
 


The roots of


are: -5, -4+3i, -4-3i, -1, -1, -1



Converting to partial fractions,
where

Thus the inverse Laplace transform of f(s) is

+



 
 
 

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A Sketch of the Derivation of the Short Cut Formulas for Obtaining Coefficients in the Partial Fractions
 
 

Case 1. (For each real and distinct root).
Suppose (-r) is an unrepeated root in the denominator, and g(s) is obtained by lumping all terms except for 1/(s+r), i.e.


       =      + [other terms]

After multiplying both sides by (s+r),
+ [other terms](s+r)
and letting s be equal to -r, only A remains on the right hand side of the equation, so
Case 2. (For each root that is repeated k times).
Suppose (-r) is a root in the denominator that is repeated k times, and g(s) is obtained by lumping all terms except for 1/(s+r)k, i.e.

=

+ [other terms]

After multiplying both sides by (s+r)k,

+ [other terms](s+r)k

Take the derivative of both sides with respect to s,

+ [other terms]k(s+r)k-1 + (s+r)k (d/ds)[other terms]

and for the second derivative,

+ [other terms]k(k-1)(s+r)k-2 + (s+r)k(d2/ds2) [other terms]

+ 2((d/ds) [other terms] )k(s+r)k-1

In general, we get for the ith derivative with respect to s,
+ [ terms containing (s+r) in the numerator ]
so by choosing s = - r at the appropriate ith derivative, we have



 
 
 
 
 

Case 3. (For each two roots that are complex conjugate pairs).
Suppose (-a+ib) and (-a-ib) are complex conjugate roots in the denominator, and g(s) is obtained by lumping all terms except for 1/[ (s+a)2 + b2 ], i.e.


+ [other terms]

Multiplying both sides by ,


  + [other terms][ (s+a)2+b2 ]


By letting Q be g(s= -a+ib), the other terms disappear, leaving only A and B on the right hand side,




 
 

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This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 1/12/2000.

     Tomas B. Co
     Associate Professor
     Department of Chemical Engineering
     Michigan Technological University
     1400 Townsend Avenue
     Houghton, MI 49931-1295

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