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Relation Theory Basic Polynomial Fourier Transform Laplace Transform Signum Function

1	Basic Polynomial
1	Relation Theory
4	Laplace Transform
4	Fourier Transform
8	Signum Function

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Heaviside Step Function

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.