Laplace transform of Dirac delta

The Dirac delta ( δ can be interpreted as a linear functionalMathworldPlanetmath, i.e. a linear mapping from a function spaceMathworldPlanetmath, consisting e.g. of certain real functions, to (or ), having the property


One may think this as the inner productMathworldPlanetmath


of a functionMathworldPlanetmath f and another “function” δ, when the well-known


is true.  Applying this to  f(t):=e-st,  one gets


i.e. the Laplace transformDlmfMathworldPlanetmath

{δ(t)}= 1. (1)

By the delay theorem, this result may be generalised to


When introducing some “nascent Dirac delta function”, for example

ηε(t):={1εfor  0tε,0for  t>ε,

as an “approximation” of Dirac delta, we obtain the Laplace transform


As the Taylor expansionMathworldPlanetmath shows, we then have

limε0+{ηε(t)}= 1,

being in accordance with (1).

Title Laplace transform of Dirac delta
Canonical name LaplaceTransformOfDiracDelta
Date of creation 2013-03-22 19:10:56
Last modified on 2013-03-22 19:10:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Result
Classification msc 46E20
Classification msc 44A10
Classification msc 34L40