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Heaviside Step Function

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The Heaviside step function, often shortened to the Heaviside step, is a mathematical function denoted H(x), or sometimes theta(x) or u(x) (Abramowitz and Stegun 1972, p. 1020), and also known as the "unit step function." The term "Heaviside step function" and its symbol can represent either a piecewise constant function or a generalized function.

HeavisideStepFunction

When defined as a piecewise constant function, the Heaviside step function is given by

H(x)={0 x<0; 1/2 x=0; 1 x>0
(1)

(Abramowitz and Stegun 1972, p. 1020; Bracewell 2000, p. 61). The plot above shows this function (left figure), and how it would appear if displayed on an oscilloscope (right figure).

When defined as a generalized function, it can be defined as a function theta(x) such that

inttheta(x)phi^'(x)dx=-phi(0)
(2)

for phi^'(x) the derivative of a sufficiently smooth function phi(x) that decays sufficiently quickly (Kanwal 1998).

The Wolfram Language represents the Heaviside generalized function as HeavisideTheta, while using UnitStep to represent the piecewise function Piecewise[{{1, x >= 0}}] (which, it should be noted, adopts the convention H(0)=1 instead of the conventional definition H(0)=1/2).

The shorthand notation

H_c(x)=H(x-c)
(3)

is sometimes also used.

The Heaviside step function is related to the boxcar function by

Pi(x)=H(x+1/2)-H(x-1/2)
(4)

and can be defined in terms of the sign function by

H(x)=1/2[1+sgn(x)].
(5)

The derivative of the step function is given by

d/(dx)H(x)=delta(x),
(6)

where delta(x) is the delta function (Bracewell 2000, p. 97).

The Heaviside step function is related to the ramp function R(x) by

R(x)=xH(x),
(7)

and to the derivative of R(x) by

d/(dx)R(x)=H(x).
(8)

The two are also connected through

R(x)=H(x)*H(x),
(9)

where * denotes convolution.

Bracewell (2000) gives many identities, some of which include the following. Letting * denote the convolution,

H(x)*f(x)=int_(-infty)^xf(x^')dx^'
(10)
H(t)*H(t)=int_(-infty)^inftyH(u)H(t-u)du
(11)
=H(0)int_0^inftyH(t-u)du
(12)
=H(0)H(t)int_0^tdu
(13)
=tH(t).
(14)

In addition,

H(ax+b)=H(x+b/a)H(a)+H(-x-b/a)H(-a)
(15)
={H(x+b/a) a>0; H(-x-b/a) a<0.
(16)
HeavisideStepFunctionLim

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

H(x)=lim_(t->0)[1/2+1/pitan^(-1)(x/t)]
(17)
=1/(sqrt(pi))lim_(t->0)int_(-x)^inftyt^(-1)e^(-u^2/t^2)du
(18)
=1/2lim_(t->0)erfc(-x/t)
(19)
=1/pilim_(t->0)int_(-infty)^xt^(-1)sinc(u/t)du
(20)
=1/pilim_(t->0)int_(-infty)^x1/usin(u/t)du
(21)
=1/2+1/pilim_(t->0)si((pix)/t)
(22)
=lim_(t->0){1/2e^(x/t) for x<=0; 1-1/2e^(-x/t) for x>=0
(23)
=lim_(t->0)1/(1+e^(-x/t))
(24)
=lim_(t->0)e^(-e^(-x/t))
(25)
=1/2lim_(t->0)[1+tanh(x/t)]
(26)
=lim_(t->0)int_(-infty)^xt^(-1)Lambda((x-1/2t)/t)dx,
(27)

where erfc(x) is the erfc function, si(x) is the sine integral, sinc(x) is the sinc function, and Lambda(x) is the one-argument triangle function. The first four of these are illustrated above for t=0.2, 0.1, and 0.01.

Of course, any monotonic function with constant unequal horizontal asymptotes is a Heaviside step function under appropriate scaling and possible reflection. The Fourier transform of the Heaviside step function is given by

F[H(x)]=int_(-infty)^inftye^(-2piikx)H(x)dx
(28)
=1/2[delta(k)-i/(pik)],
(29)

where delta(k) is the delta function.

See also

Absolute Value, Boxcar Function, Delta Function, Fourier Transform--Heaviside Step Function, Ramp Function, Rectangle Function, Sigmoid Function, Sign, Square Wave, Triangle Function

Related Wolfram sites

https://functions.wolfram.com/GeneralizedFunctions/UnitStep/, https://functions.wolfram.com/GeneralizedFunctions/UnitStep2/

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

  • rectangle function
  • (unit step function (y-x) + |x +y| <= 2 )
  • unit step function of 2pi

References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Bracewell, R. "Heaviside's Unit Step Function, H(x)." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 61-65, 2000.Kanwal, R. P. Generalized Functions: Theory and Technique, 2nd ed. Boston, MA: Birkhäuser, 1998.Spanier, J. and Oldham, K. B. "The Unit-Step u(x-a) and Related Functions." Ch. 8 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 63-69, 1987.

Referenced on Wolfram|Alpha

Heaviside Step Function

Cite this as:

Weisstein, Eric W. "Heaviside Step Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HeavisideStepFunction.html

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