Heaviside Expansion Formula/Also presented as

Heaviside Expansion Formula: Also known as

Heaviside Expansion Formula 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:

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


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: $32.24$
  • 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: $33.24.$
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