Heaviside step function

The Heaviside step function is the functionMathworldPlanetmath H: defined as

H(x) = {0whenx<0,1/2whenx=0,1whenx>0.

Here, there are many conventions for the value at x=0. The motivation for setting H(0)=1/2 is that we can then write H as a function of the signum function (see this page (http://planetmath.org/SignumFunction)). In applications, such as the Laplace transformDlmfMathworldPlanetmath, where the Heaviside function is used extensively, the value of H(0) is irrelevant. The Fourier transformDlmfMathworldPlanetmath of heaviside function is


where δ denotes the Dirac delta centered at 0. The function is named after Oliver Heaviside (1850-1925) [1]. However, the function was already used by Cauchy[2], who defined the function as


and called it a coefficient limitateur [3].


  • 1 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Heav iside.htmlOliver Heaviside.
  • 2 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Cauc hy.htmlAugustin Louis Cauchy.
  • 3 R.F. Hoskins, Generalised functions, Ellis Horwood Series: Mathematics and its applications, John Wiley & Sons, 1979.
Title Heaviside step function
Canonical name HeavisideStepFunction
Date of creation 2013-03-22 13:46:14
Last modified on 2013-03-22 13:46:14
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Definition
Classification msc 30-00
Classification msc 26A06
Synonym Heaviside function
Related topic SignumFunction
Related topic DelayTheorem
Related topic TelegraphEquation