differentiation of Laplace transform with respect to parameter

We use the curved from the Laplace-transformed functionsMathworldPlanetmath to the corresponding initial functions.



then one can differentiate both functions with respect to the parametre x:

fx(t,x)Fx(s,x) (1)

(1) may be written also as

{xf(t,x)}=x{f(t,x)}. (2)

Proof.  We differentiate partially both sides of the defining equation


on the right hand side under the integration sign (http://planetmath.org/differentiationundertheintegralsign), getting

Fx(s,x)=0e-stfx(t,x)𝑑x, (3)

which means same as (1).  Q.E.D.

Example.  If the rule


is differentiated with respect to a, the result is



  • 1 K. Väisälä: Laplace-muunnos.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
Title differentiation of Laplace transform with respect to parameter
Canonical name DifferentiationOfLaplaceTransformWithRespectToParameter
Date of creation 2014-03-09 12:52:11
Last modified on 2014-03-09 12:52:11
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 2
Author pahio (2872)
Entry type Theorem
Classification msc 44A10