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 $x=0$. 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

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

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 HeavisideStepFunction 2013-03-22 13:46:14 2013-03-22 13:46:14 Koro (127) Koro (127) 8 Koro (127) Definition msc 30-00 msc 26A06 Heaviside function SignumFunction DelayTheorem TelegraphEquation