# 2.3: Partial-Fraction Expansion

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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.

This page titled 2.3: Partial-Fraction Expansion is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.