10. Partial Fraction Expansion

Not every \(F(s)\) we encounter is in the Laplace table. Partial fractions is a method for re-writing \(F(s)\) in a form suitable for the use of the table.

Rational Functions

A rational function is one that is the ratio of two polynomials. For example \[\frac{s + 1}{s^2 + 7s + 9} \qquad \text{ and } \qquad \frac{s^2 + 7s + 9}{s + 1}\] are both rational functions.

A rational function is called proper if the degree of the numerator is strictly smaller than the degree of the denominator; in the examples above, the first is proper while the second is not.

The partial fraction decomposition only applies to proper functions.

In general, if \(P(s)/Q(s)\) is a proper rational function and \(Q(s)\) factors into distinct linear factors \(Q(s) = (s - a_1)(s - a_2)\cdots(s - a_n)\) then \[\frac{P(s)}{Q(s)} = \frac{A_1}{s-a_1} + \frac{A_2}{s-a_2}+\cdots+\frac{A_n}{s - a_n}\]

Use partial fractions to find \(\mathcal{L}^{-1}\left(\dfrac{3}{s^3 - 3s^2- s + 3}\right)\).