Heaviside step function
The Heaviside step function is the function H:ℝ→ℝ 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 transform, where the Heaviside function is used extensively,
the value of H(0) is irrelevant.
The Fourier transform
of heaviside function is
ℱ0H(t)=12(δ(t)-iπt) |
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
u(t)=12(1+t/√t2) |
and called it a coefficient limitateur [3].
References
- 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 |