Heaviside formula

Let P(s) and Q(s) be polynomialsMathworldPlanetmathPlanetmath with the degree of the former less than the degree of the latter.

  • If all complex zeroes (http://planetmath.org/Zero) a1,a2,,an of Q(s) are simple, then

    -1{P(s)Q(s)}=j=1nP(aj)Q(aj)eajt. (1)
  • If the different zeroes a1,a2,,an of Q(s) have the multiplicities m1,m2,,mn, respectively, we denote  Fj(s):=(s-aj)mjP(s)/Q(s);  then

    -1{P(s)Q(s)}=j=1neajtk=0mj-1Fj(k)(aj)tmj-1-kk!(mj-1-k)!. (2)

A special case of the Heaviside formula (1) is

-1{Q(s)Q(s)}=j=1neajt. (3)

Example.  Since the zeros of the binomial s4+4a4 are  s=(±1±i)a,  we can calculate by (3) as follows:


Proof of (1).  Without hurting the generality, we can suppose that Q(s) is monic.  Therefore


For  j=1,2,,n,  denoting


one has  Qj(aj)0.  We have a partial fraction expansion of the form

P(s)Q(s)=C1s-a1+C2s-a2++Cns-an (4)

with constants Cj.  According to the linearity and the formula 1 of the parent entry (http://planetmath.org/LaplaceTransform), one gets

-1{P(s)Q(s)}=j=1nCjeajt. (5)

For determining the constants Cj, multiply (3) by s-aj.  It yields


Setting to this identityPlanetmathPlanetmaths:=aj  gives the value

Cj=P(aj)Qj(aj). (6)

But since  Q(s)=dds((s-aj)Qj(s))=Qj(s)+(s-aj)Qj(s),  we see that  Q(aj)=Qj(aj);  thus the equation (5) may be written

Cj=P(aj)Q(aj). (7)

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


  • 1 K. Väisälä: Laplace-muunnos.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
Title Heaviside formula
Canonical name HeavisideFormula
Date of creation 2014-03-19 9:14:46
Last modified on 2014-03-19 9:14:46
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Topic
Classification msc 44A10
Synonym Heaviside expansion formula
Synonym inverse Laplace transform of rational function
Related topic HyperbolicFunctions
Related topic ComplexSineAndCosine