Laplace transform of logarithm

Theorem.  The Laplace transformDlmfMathworldPlanetmath of the natural logarithmMathworldPlanetmathPlanetmathPlanetmath functionMathworldPlanetmath is


where Γ is Euler’s gamma functionDlmfDlmfMathworldPlanetmath.

Proof.  We use the Laplace transform of the power functionDlmfDlmfPlanetmath (


by differentiating it with respect to the parametre a:


Setting here  a=0,  we obtain



Note.  The number Γ(1) is equal the of the Euler–Mascheroni constant (, 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