Heaviside step function: Difference between revisions

Revision as of 15:39, 18 August 2024

Revision as of 06:29, 7 September 2024

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| fields_of_application = Operational calculus

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The '''Heaviside step function''', or the '''unit step function''', usually denoted by {{mvar|H}} or {{mvar|θ}} (but sometimes {{mvar|u}}, {{math|'''1'''}} or {{math|{{not a typo|𝟙}}}}), is a [[step function]] named after [[Oliver Heaviside]], the value of which is [[0 (number)|zero]] for negative arguments and [[1 (number)|one]] for positive arguments. Different conventions concerning the value {{math|''H''(0)}} are in use. It is an example of the general class of step functions, all of which can be represented as [[linear combination]]s of translations of this one.

The function was originally developed in [[operational calculus]] for the solution of [[differential equation]]s, where it represents a signal that switches on at a specified time and stays switched on indefinitely. [[Oliver Heaviside]], who developed the operational calculus as a tool in the analysis of telegraphic communications, and represented the function as {{math|'''1'''}}.

==Formulation==

Taking the convention that {{math|''H''(0) {{=}} 1}}, the Heaviside function may be defined as:

* a [[piecewise function]]: <math display="block">H(x) := \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}</math>

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* the derivative of the [[ramp function]]: <math display="block">H(x) := \frac{d}{dx} \max \{ x, 0 \}\quad \mbox{for } x \ne 0</math>

TheFurthermore, the Heaviside step function can be represented as a [[hyperfunction]] as

<math display="block">H(x) = \left(1-\frac{1}{2\pi i}\log z,\ -\frac{1}{2\pi i}\log z\right).</math>

where {{math|log ''z''}} is the [[Complex logarithm#Principal value|principal value of the complex logarithm]] of {{mvar|z}}.

It can also be expressed for {{math|''x'' ≠ 0}} in terms of the [[absolute value]] function as

<math display="block"> H(x) = \frac{x + |x|}{2x} \,.</math>

==Relationship with Diract delta==

The [[Dirac delta function]] is the [[derivative]] of the Heaviside function:

<math display="block">\delta(x)= \frac{d}{dx} H(x).</math>

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although this expansion may not hold (or even make sense) for {{math|''x'' {{=}} 0}}, depending on which formalism one uses to give meaning to integrals involving {{mvar|δ}}. In this context, the Heaviside function is the [[cumulative distribution function]] of a [[random variable]] which is [[almost surely]] 0. (See [[Constant random variable]].)

== Analytic approximations ==

Approximations to the Heaviside step function are of use in [[biochemistry]] and [[neuroscience]], where [[logistic function|logistic]] approximations of step functions (such as the [[Hill equation (biochemistry)|Hill]] and the [[Michaelis–Menten kinetics|Michaelis–Menten equations]]) may be used to approximate binary cellular switches in response to chemical signals.

== Analytic approximations ==

[[File:Step function approximation.png|alt=A set of functions that successively approach the step function|thumb|500x500px|<math>\tfrac{1}{2} + \tfrac{1}{2} \tanh(kx) = \frac{1}{1+e^{-2kx}}</math><br>approaches the step function as {{math|''k'' → ∞}}.]]

For a [[Smooth function|smooth]] approximation to the step function, one can use the [[logistic function]]

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When the bilateral transform is used, the integral can be split in two parts and the result will be the same.

==Other expressions==

The Heaviside step function can be represented as a [[hyperfunction]] as

<math display="block">H(x) = \left(1-\frac{1}{2\pi i}\log z,\ -\frac{1}{2\pi i}\log z\right).</math>

where {{math|log ''z''}} is the [[Complex logarithm#Principal value|principal value of the complex logarithm]] of {{mvar|z}}.

It can also be expressed for {{math|''x'' ≠ 0}} in terms of the [[absolute value]] function as

<math display="block"> H(x) = \frac{x + |x|}{2x} \,.</math>

==See also==