The Heaviside step function is a mathematical function denoted , or sometimes or (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.
When defined as a piecewise constant function, the Heaviside step function is given by
(1)
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(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 such that
(2)
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for the derivative of a sufficiently smooth function 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 instead of the conventional definition ).
The shorthand notation
(3)
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is sometimes also used.
The Heaviside step function is related to the boxcar function by
(4)
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and can be defined in terms of the sign function by
(5)
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The derivative of the step function is given by
(6)
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where is the delta function (Bracewell 2000, p. 97).
The Heaviside step function is related to the ramp function by
(7)
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and to the derivative of by
(8)
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The two are also connected through
(9)
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where denotes convolution.
Bracewell (2000) gives many identities, some of which include the following. Letting denote the convolution,
(10)
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(11)
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(12)
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(13)
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(14)
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In addition,
(15)
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(16)
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The Heaviside step function can be defined by the following limits,
(17)
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(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(25)
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(26)
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(27)
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where is the erfc function, is the sine integral, is the sinc function, and is the one-argument triangle function. The first four of these are illustrated above for , 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
(28)
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(29)
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where is the delta function.