Laplace transform of logarithm


Theorem.  The Laplace transformDlmfMathworldPlanetmath of the natural logarithmMathworldPlanetmathPlanetmathPlanetmath functionMathworldPlanetmath is

{lnt}=Γ(1)-lnss

where Γ is Euler’s gamma functionDlmfDlmfMathworldPlanetmath.

Proof.  We use the Laplace transform of the power functionDlmfDlmfPlanetmath (http://planetmath.org/LaplaceTransformOfPowerFunction)

0e-stta𝑑t=Γ(a+1)sa+1

by differentiating it with respect to the parametre a:

0e-sttalntdt=Γ(a+1)sa+1-Γ(a+1)sa+1lns(sa+1)2=Γ(a+1)-Γ(a+1)lnssa+1

Setting here  a=0,  we obtain

{lnt}=0e-stlntdt=Γ(1)-1lnss,

Q.E.D.

Note.  The number Γ(1) is equal the of the Euler–Mascheroni constant (http://planetmath.org/EulersConstant), as is seen in the entry digamma and polygamma functions.

Title Laplace transform of logarithm
Canonical name LaplaceTransformOfLogarithm
Date of creation 2013-03-22 18:26:01
Last modified on 2013-03-22 18:26:01
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 44A10
Synonym Laplace transform of logarithm function
Related topic PowerFunction