Heaviside step function
The Heaviside step function is the function^{} $H:\mathbb{R}\to \mathbb{R}$ defined as
$H(x)$ | $=$ | $$ |
Here, there are many conventions for the value at $x=0$. The motivation for setting $H(0)=1/2$ is that we can then write $H$ as a function of the signum function (see this page (http://planetmath.org/SignumFunction)). In applications, such as the Laplace transform^{}, where the Heaviside function is used extensively, the value of $H(0)$ is irrelevant. The Fourier transform^{} of heaviside function is
$${\mathcal{F}}_{0}H(t)=\frac{1}{2}\left(\delta (t)-\frac{i}{\pi t}\right)$$ |
where $\delta $ denotes the Dirac delta centered at $0$. The function is named after Oliver Heaviside (1850-1925) [1]. However, the function was already used by Cauchy[2], who defined the function as
$$u(t)=\frac{1}{2}\left(1+t/\sqrt{{t}^{2}}\right)$$ |
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 |