PlanetMath: Heaviside step function

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Heaviside step function

(Definition)

The Heaviside step function is the function $H:\mathbb{R}\to \mathbb{R}$ defined as

$\displaystyle H(x)$

$\displaystyle =$

$\displaystyle \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \ 1/2 & \mbox{when}\,\, x= 0,\ 1 & \mbox{when}\,\, x> 0.\ \end{array} \right.$

Here, there are many conventions for the value at

. The motivation for setting

is that we can then write

as a function of the signum function (see this page). In applications, such as the Laplace transform, where the Heaviside function is used extensively, the value of

is irrelevant. The Fourier transform of heaviside function is

$\displaystyle \mathcal{F}_0 H(t)=\frac{1}{2}\left(\delta(t)-\frac{i}{\pi t}\right)$

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

$\displaystyle u(t) = \frac{1}{2}\big( 1 + t/\sqrt{t^2}\big)$

and called it a coefficient limitateur [3].

Bibliography

1: The MacTutor History of Mathematics archive, Oliver Heaviside.
2: The MacTutor History of Mathematics archive, Augustin Louis Cauchy.
3: R.F. Hoskins, Generalised functions, Ellis Horwood Series: Mathematics and its applications, John Wiley & Sons, 1979.

"Heaviside step function" is owned by Koro. [ full author list (2) | owner history (1) ]

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Other names:

Heaviside function

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Cross-references: coefficient, Fourier transform, Laplace transform, applications, signum function, function
There are 2 references to this entry.

This is version 5 of Heaviside step function, born on 2003-07-18, modified 2008-05-15.
Object id is 4476, canonical name is HeavisideStepFunction.
Accessed 19978 times total.

Classification:

AMS MSC:	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	30-00 (Functions of a complex variable :: General reference works )

Pending Errata and Addenda

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