integration of Laplace transform with respect to parameter


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

If

f(t,x)F(s,x),

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

abf(t,x)𝑑x0(e-stabf(s,x)𝑑x)𝑑t=0(abe-stf(s,x)𝑑x)𝑑t.

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

abf(t,x)𝑑xab(0e-stf(s,x)𝑑t)𝑑x=abF(s,t)𝑑x,

Q.E.D.

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
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