Heaviside Expansion Formula

Theorem

Let $P, Q$ be polynomials with coefficients in $\C$.

Let $\deg Q \ge \deg P + 1$.

Let $\map Q z$ have a simple zero for $z \in X$.

Let $\map {\laptrans f} z = \dfrac {\map P z} {\map Q z}$.

Then:

$\ds \map f t = \sum_{z \mathop \in X} e^{z t} \frac {\map P z} {\map {Q'} z}$

Proof

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Also presented as

can also be found in the form:

Let $P, Q$ be polynomials with coefficients in $\C$.

Let $\deg Q \ge \deg P + 1$.

Let $\map Q s = \ds \prod_{k \mathop = 1}^n \paren {s - a_k}$ where all $a_1, a_2, \ldots, a_n$ are distinct.

Let $\map {\laptrans f} s = \dfrac {\map P s} {\map Q s}$.

Then:

$\ds \map f t = \sum_{z \mathop = 1}^n \frac {\map P {a_k} } {\map {Q'} {a_k} } e^{a_k t}$

Source of Name

This entry was named for Oliver Heaviside.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text A$: Table of General Properties of Laplace Transforms: $22.$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Laplace Transforms: Table of General Properties of Laplace Transforms: $32.24$
• 1999: Jerrold E. Marsden and Michael J. Hoffman: Basic Complex Analysis (3rd ed.): $8.2.3$: Heaviside Expansion Formula
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of General Properties of Laplace Transforms: $33.24.$