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Heaviside step function (Definition)

The Heaviside step function is the function $H:\sR\to \sR$ defined as \begin{eqnarray*} H(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \\ 1/2 & \mbox{when}\,\, x= 0,\\ 1 & \mbox{when}\,\, x> 0.\\ \end{array}. 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). 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 $$ u(t) = \frac{1}{2}\big( 1 + t/\sqrt{t^2}\big)$$ 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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See Also: signum function, delay theorem, telegraph equation

Other names:  Heaviside function
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Cross-references: coefficient, Fourier transform, Laplace transform, applications, signum function, function
There are 3 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 22925 times total.

Classification:
AMS MSC26A06 (Real functions :: Functions of one variable :: One-variable calculus)
 30-00 (Functions of a complex variable :: General reference works )

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