Laplace transform of derivative

Theorem.  If the real functiontf(t)  and its derivative are Laplace-transformable and f is continuousMathworldPlanetmath for  t>0,  then

{f(t)}=sF(s)-limt0+f(t). (1)

Proof.  By the definition of Laplace transformDlmfMathworldPlanetmath and using integration by parts, the left hand side of (1) may be written


The Laplace-transformability of f implies that e-stf(t) tends to zero as t increases boundlessly.  Thus the last expression leads to the right hand side of (1).

Title Laplace transform of derivative
Canonical name LaplaceTransformOfDerivative
Date of creation 2013-03-22 18:24:54
Last modified on 2013-03-22 18:24:54
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 44A10
Related topic SubstitutionNotation