Converting to Partial Fractions

Short Cut Methods
Let 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  To top of page

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 + To top of page

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,   To top of page

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