# integration of Laplace transform with respect to parameter

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

If

 $f(t,\,x)\;\curvearrowleft\;F(s,\,x),$

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:

 $\displaystyle\int_{a}^{b}\!f(t,\,x)\,dx\;\curvearrowleft\;\int_{a}^{b}\!F(s,\,% x)\,dx$ (1)

(1) may be written as

 $\displaystyle\mathcal{L}\{\int_{a}^{b}\!f(t,\,x)\,dx\}\;=\;\int_{a}^{b}\!% \mathcal{L}\{f(t,\,x)\}\,dx.$ (2)

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

 $\int_{a}^{b}\!f(t,\,x)\,dx\;\curvearrowleft\;\int_{0}^{\infty}\left(e^{-st}% \int_{a}^{b}\!f(s,\,x)\,dx\right)dt\;=\;\int_{0}^{\infty}\left(\int_{a}^{b}\!e% ^{-st}f(s,\,x)\,dx\right)dt.$

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

 $\int_{a}^{b}\!f(t,\,x)\,dx\;\curvearrowleft\;\int_{a}^{b}\left(\int_{0}^{% \infty}\!e^{-st}f(s,\,x)\,dt\right)dx\;=\;\int_{a}^{b}\!F(s,\,t)\,dx,$

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 Related topic TermwiseDifferentiation Related topic MethodsOfEvaluatingImproperIntegrals Related topic UsingConvolutionToFindLaplaceTransform Related topic RelativeOfCosineIntegral Related topic RelativeOfExponentialIntegral