Heaviside Step Function

$\begin{figure}\begin{center}\BoxedEPSF{Heaviside.epsf scaled 700}\end{center}\end{figure}$

A discontinuous ``step'' function, also called the Unit Step, and defined by

$\begin{displaymath} H(x) = \cases{ 0 & $x < 0$\cr {\textstyle{1\over 2}}& $x = 0$\cr 1 & $x > 0$.\cr} \end{displaymath}$

(1)

It is related to the Boxcar Function. The Derivative is given by

$\begin{displaymath} {d\over dx} H(x) = \delta(x), \end{displaymath}$

(2)

where $\delta(x)$ is the Delta Function, and the step function is related to the Ramp Function

$\begin{displaymath} {d\over dx} R(x)=-H(x). \end{displaymath}$

(3)

Bracewell (1965) gives many identities, some of which include the following. Letting denote the Convolution,

$\begin{displaymath} H(x)*f(x) = \int^x_{-\infty} f(x')\,dx' \end{displaymath}$

(4)

$\displaystyle H(t)* H(t)$	$\textstyle =$	$\displaystyle \int_{-\infty}^\infty H(u)H(t-u)\,du$	(5)
	$\textstyle =$	$\displaystyle H(0)\int^\infty_0 H(t-u)\,du$
	$\textstyle =$	$\displaystyle H(0)H(t) \int^t_0\,du= tH(t).$	(6)

Additional identities are

$\begin{displaymath} H(x)H(y) = \cases{ H(x) & $x > y$\cr H(y) & $x < y$\cr} \end{displaymath}$

(7)

$\displaystyle H(ax+b)$	$\textstyle =$	$\displaystyle H\left({x+{b\over a}}\right)H(a)+H\left({-x-{b\over a}}\right)H(-a)$
	$\textstyle =$	$\displaystyle \left\{\begin{array}{ll} H\left({x + {b\over a}}\right)& \mbox{$a > 0$}\\ H\left({-x -{b\over a}}\right)& \mbox{$a < 0$.}\end{array}\right.$	(8)

The step function obeys the integral identities

$\displaystyle \int^b_{-a} H(u-u_0)f(u)\,du$	$\textstyle =$	$\displaystyle H(u_0) \int^b_{u_0} f(u)\,du$	(9)
$\displaystyle \int^b_{-a} H(u_1-u)f(u)\,du$	$\textstyle =$	$\displaystyle H(u_1) \int^{u_1}_{-a} f(u)\,du$	(10)

$\begin{displaymath} \int^b_{-a} H(u-u_0)H(u_1-u)f(u)\,du = H(u_0)H(u_1)\int^{u_1}_{u_0} f(u)\,du. \end{displaymath}$

(11)

The Heaviside step function can be defined by the following limits,

$\displaystyle H(x)$	$\textstyle =$	$\displaystyle \lim_{t\to 0}\left[{{\textstyle{1\over 2}}+ {1\over\pi} \tan^{-1}\left({x\over t}\right)}\right]$	(12)
	$\textstyle =$	$\displaystyle {1\over 2} \lim_{t\to 0} \mathop{\rm erfc}\nolimits \left({-{x\over t}}\right)$	(13)
	$\textstyle =$	$\displaystyle {1\over\sqrt{\pi}} \lim_{t\to 0} \int^\infty_{-x} t^{-1}e^{-u^2/t^2}\,du$	(14)
	$\textstyle =$	$\displaystyle {1\over 2} + {1\over\pi} \lim_{t\to 0} \mathop{\rm si}\nolimits \left({\pi x\over t}\right)$	(15)
	$\textstyle =$	$\displaystyle {1\over\pi} \lim_{t\to 0} \int^x_{-\infty} t^{-1}\mathop{\rm sinc}\nolimits \left({u\over t}\right)\,du$	(16)
	$\textstyle =$	$\displaystyle \lim_{t\to 0} \left\{\begin{array}{ll} {\textstyle{1\over 2}}e^{x... ...$}\\ 1-{\textstyle{1\over 2}}e^{-x/t} & \mbox{for $x\geq 0$}\end{array}\right.$	(17)
	$\textstyle =$	$\displaystyle \lim_{t\to 0} \int^x_{-\infty}t^{-1}\Lambda\left({x-{\textstyle{1\over 2}}t\over t}\right)\,dx,$	(18)

where $\mathop{\rm erfc}\nolimits (x)$ is the Erfc function, $\mathop{\rm si}\nolimits (x)$ is the Sine Integral, $\mathop{\rm sinc}\nolimits x$ is the Sinc Function, and $\Lambda(x)$ is the one-argument Triangle Function and

The Fourier Transform of the Heaviside step function is given by

$\begin{displaymath} {\mathcal F}[H(x)] = \int_{-\infty}^\infty e^{-2\pi ikx}H(x)\,dx = {1\over 2}\left[{\delta(k)-{i\over \pi k}}\right], \end{displaymath}$

(19)

where $\delta(k)$ is the Delta Function.

References

Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, 1965.

Spanier, J. and Oldham, K. B. ``The Unit-Step and Related Functions.'' Ch. 8 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 63-69, 1987.