Laplace transform of logarithm
Theorem. The Laplace transform of the natural logarithm
function
is
ℒ{lnt}=Γ′(1)-lnss |
where Γ is Euler’s gamma function.
Proof. We use the Laplace transform of the power function (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
ℒ{lnt}=∫∞0e-stlntdt=Γ′(1)-1⋅lnss, |
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 |