PlanetPhysics/Heaviside Formula

Let and be polynomials with the degree of the former less than the degree of the latter.

  • If all complex zeroes of are simple, then

  • If the different zeroes of have the multiplicities , respectively, we denote\, ;\, then

A special case of the Heaviside formula (1) is Failed to parse (syntax error): {\displaystyle \mathcal{L}^{-1}\left\{\frac{Q'(s)}{Q(s)}\right\} \;=\; \sum_{j=1}^ne^{a_jt}.\\}

Example. \, Since the zeros of the binomial are\, ,\, we obtain \\

Proof of (1). \, Without hurting the generality, we can suppose that is monic.\, Therefore For\, ,\, denoting one has\, .\, We have a partial fraction expansion of the form

with constants .\, According to the linearity and the formula 1 of the parent entry, one gets

For determining the constants , multiply (3) by .\, It yields Setting to this identity \,\, gives the value

But since\, ,\, we see that\, ;\, thus the equation (5) may be written

The values (6) in (4) produce the formula (1).

All SourcesEdit

[1]

ReferencesEdit

  1. {\sc K. V\"ais\"al\"a:} Laplace-muunnos .\, Handout Nr. 163.\quad Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).