2.3: Partial-Fraction Expansion

Let us examine in more detail the justification for the form of the partial-fraction expansion presented in Equation 2.2.15. As an example, consider the following sum of three fractional terms:

\[\begin{align}
F_{3}(s)&=\sum_{k=1}^{3} \frac{C_{k}}{s-p_{k}}=\frac{C_{1}}{s-p_{1}}+\frac{C_{2}}{s-p_{2}}+\frac{C_{3}}{s-p_{3}}\\[4pt]
&=\frac{C_{1}\left(s-p_{2}\right)\left(s-p_{3}\right)+C_{2}\left(s-p_{1}\right)\left(s-p_{3}\right)+C_{3}\left(s-p_{1}\right)\left(s-p_{2}\right)}{\left(s-p_{1}\right)\left(s-p_{2}\right)\left(s-p_{3}\right)}\label{eqn:2.23}
\end{align} \nonumber \]

In the combination of the three fractional terms into a single fraction, the denominator is a cubic polynomial (\(n = 3\)), and the degree of the numerator polynomial is, at most, \(m = 2\). Depending upon the values of constants \(C_k\), the degree of the numerator polynomial can be from 0 to 2, so that 0 \(\leq\) \(m\) < \(n\).

Next, let us generalize the observation of the previous paragraph by considering a general sum of fractional terms in the form

\[F_{n}(s)=\sum_{k=1}^{n} \frac{C_{k}}{s-p_{k}}\label{eqn:2.24} \]

If we combine all \(n\) of these terms into a single ratio using the traditional algebraic method illustrated in Equation \(\ref{eqn:2.23}\), then the result will have the following general form:

\[F_{n}(s)=\frac{c_{1} s^{m}+c_{2} s^{m-1}+\ldots+c_{m+1}}{\left(s-p_{1}\right)\left(s-p_{2}\right) \cdots\left(s-p_{n}\right)} \nonumber \]

The denominator is a polynomial of degree \(n\), and the numerator is a polynomial of, at most, degree \(m\) = (\(n\) − 1). We therefore can conclude the following: a ratio of polynomials, in which the numerator has a lower degree than that of the denominator, can usually be expanded into the simple partial-fraction form Equation \(\ref{eqn:2.24}\). In other words, provided that 0 \(\leq\) \(m\) < \(n\), we can usually find finite residues \(C_k\) in the partial-fraction expansion:

\[F_{n}(s)=\frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n+1}}=\frac{b_{1}\left(s-z_{1}\right)\left(s-z_{2}\right) \cdots\left(s-z_{m}\right)}{a_{1}\left(s-p_{1}\right)\left(s-p_{2}\right) \cdots\left(s-p_{n}\right)}=\sum_{k=1}^{n} \frac{C_{k}}{s-p_{k}}\label{eqn:2.25a} \]

with the residues given by the labor-saving method as¹

\[C_{k}=\left[\left(s-p_{k}\right) F_{n}(s)\right]_{s=p_{k}}, k=1,2, \dots, n\label{eqn:2.25b} \]

Consider, for example, part of the partial-fraction expansion of a quadratic divided by a cubic:

\[F_{3}(s)=\frac{b_{1} s^{2}+b_{2} s+b_{3}}{\left(s-p_{1}\right)\left(s-p_{2}\right)\left(s-p_{3}\right)}=\frac{C_{1}}{s-p_{1}}+\frac{C_{2}}{s-p_{2}}+\frac{C_{3}}{s-p_{3}} \nonumber \]

Using Equation \(\ref{eqn:2.25b}\), to determine, for example, residue \(C_1\) gives:

\[C_{1}=\left[\left(s-p_{1}\right) F_{3}(s)\right]_{s=p_{1}}=\frac{b_{1} p_{1}^{2}+b_{2} p_{1}+b_{3}}{\left(p_{1}-p_{2}\right)\left(p_{1}-p_{3}\right)} \nonumber \]

This equation for \(C_1\) reveals an exception to the rule: this equation clearly would not be valid if the denominator polynomial were to have repeated roots, \(p_{1}=p_{2}\) or \(p_{1}=p_{3}\); in that case, a form different than Equations \(\ref{eqn:2.25a}\) and \(\ref{eqn:2.25b}\) would be appropriate (Ogata, 1998, pp. 33-34). That is a special case which we shall address in this book only as the need arises.

Finally, observe that it is easy to check the validity/correctness of a partial-fraction expansion after we have solved for residues \(C_k\). Simply combine the individual fractions into a single ratio, as is illustrated in Equation \(\ref{eqn:2.23}\); the resulting ratio should equal the original ratio of polynomials that we expanded into partial fractions.

¹An interesting alternative form of Equation \(\ref{eqn:2.25b}\) is developed in Appendix A, Section A-2.