integration of Laplace transform with respect to parameter

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



then one can integrate both functions with respect to the parametre x between the same which may be also infinite provided that the integrals converge:

abf(t,x)𝑑xabF(s,x)𝑑x (1)

(1) may be written as

{abf(t,x)𝑑x}=ab{f(t,x)}𝑑x. (2)

Proof.  Using the definition of the Laplace transformDlmfMathworldPlanetmath, we can write


We change the of integration in the last double integral and use again the definition, obtaining



Title integration of Laplace transform with respect to parameter
Canonical name IntegrationOfLaplaceTransformWithRespectToParameter
Date of creation 2013-03-22 18:44:47
Last modified on 2013-03-22 18:44:47
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 44A10
Related topic TableOfLaplaceTransforms
Related topic TermwiseDifferentiation
Related topic MethodsOfEvaluatingImproperIntegrals
Related topic UsingConvolutionToFindLaplaceTransform
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