Heaviside step function: Difference between revisions

Revision as of 05:34, 5 April 2022

Revision as of 12:34, 12 June 2022

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== Analytic approximations ==

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

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

<math display="block">H(x) \approx \tfrac{1}{2} + \tfrac{1}{2}\tanh kx = \frac{1}{1+e^{-2kx}},</math>

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There exist various reasons for choosing a particular value.

* {{math|''H''(0) {{=}} {{sfrac|1|2}}}} is often used since the [[graph of a function|graph]] then has rotational symmetry; put another way, {{math|''H'' − {{sfrac|1|2}}}} is then an [[odd function]]. In this case the following relation with the [[sign function]] holds for all {{mvar|x}}: <math display="block"> H(x) = \tfrac12(1 + \tfrac12\sgn (x).</math>

* {{math|''H''(0) {{=}} 1}} is used when {{mvar|H}} needs to be [[right-continuous]]. For instance [[cumulative distribution function]]s are usually taken to be right continuous, as are functions integrated against in [[Lebesgue–Stieltjes integration]]. In this case {{mvar|H}} is the [[indicator function]] of a [[closed set|closed]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{[0,\infty)}(x).</math> The corresponding probability distribution is the [[degenerate distribution]].

* {{math|''H''(0) {{=}} 0}} is used when {{mvar|H}} needs to be [[left-continuous]]. In this case {{mvar|H}} is an indicator function of an [[open set|open]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{(0,\infty)}(x).</math>

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==Discrete form==

An alternative form of the unit step, defined instead as a function <{{math>|''H'' : \mathbb{Z}ℤ \to→ \mathbb{Rℝ}}</math> (that is, taking in a discrete variable {{mvar|n}}), is:

<math display="block">H[n]=\begin{cases} 0, & n < 0, \\ 1, & n \ge 0, \end{cases} </math>

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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)</math>''}} is the [[Complex logarithm#Principal value|principal value of the complex logarithm]] of {{mvar|z}}.

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

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