inverse Laplace transform of derivatives

It may be shown that the Laplace transformDlmfMathworldPlanetmathF(s)=0e-stf(t)𝑑t  is always differentiableMathworldPlanetmathPlanetmath and that its derivative can be formed by differentiating under the integral sign (, i.e. one has


This gives the rule

-1{F(s)}=-tf(t). (1)

Applying (1) to F(s) instead of F(s) gives


Continuing this way we can obtain the general rule

-1{F(n)(s)}=(-1)ntnf(t), (2)

or equivalently

{tnf(t)}=(-1)ndn{f(t)}dsn, (3)

for any  n=1, 2, 3, (and of course for  n=0).

Example.  Let’s find the Laplace transform of the first kind and 0th Bessel functionDlmfMathworldPlanetmathPlanetmath


which is the solution y(t) of the Bessel’s equation

ty′′(t)+y(t)+ty(t)=0 (4)

satisfying the initial conditionMathworldPlanetmathy(0)=1.  The equation implies that  y(0)=0.

By (3), the Laplace transform of the differential equationMathworldPlanetmath (4) is


Using here twice the rule 5 in the parent ( entry gives us


which is simplified to


i.e. to


Integrating this gives




The initial condition enables to justify that the integration constant C must be 1.  Thus we have the result



  • 1 K. Väisälä: Laplace-muunnos.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
Title inverse Laplace transform of derivatives
Canonical name InverseLaplaceTransformOfDerivatives
Date of creation 2013-03-22 16:46:27
Last modified on 2013-03-22 16:46:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Derivation
Classification msc 44A10
Synonym differentiation of Laplace transform
Related topic MellinsInverseFormula
Related topic SeparationOfVariables
Related topic KalleVaisala
Related topic TableOfLaplaceTransforms