Heaviside step function: Difference between revisions

Revision as of 08:48, 6 August 2025

Latest revision as of 08:49, 6 August 2025

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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>where a larger {{mvar|k}} corresponds to a sharper transition at {{math|''x'' {{=}} 0}}.

[[File:StepIf functionwe approximation.pngtake {{math|alt''H''(0) {{=A}} set of{{sfrac|1|2}}}}, functionsequality thatholds successively approachin the step function|thumb|500x500px|limit:<math display="block">H(x)=\tfraclim_{1}{2}k +\to \infty}\tfrac{1}{2} (1+\tanh( kx) =\lim_{k \to \infty}\frac{1}{1+e^{-2kx}}.</math><br>approaches the step function as {{math|''k'' → ∞}}.|left]]

If[[File:Step wefunction take {{mathapproximation.png|''H''(0) {{alt=}}A {{sfrac|1|2}}}},set of equalityfunctions holdsthat insuccessively approach the limit:step function|thumb|500x500px|<math display="block">H(x)=\lim_tfrac{k1}{2} \to+ \infty}\tfrac{1}{2}(1+ \tanh (kx) =\lim_{k \to \infty}\frac{1}{1+e^{-2kx}}.</math><br>approaches the step function as {{math|''k'' → ∞}}.|none]]There are [[Sigmoid function#Examples|many other smooth, analytic approximations]] to the step function.<ref>{{MathWorld | urlname=HeavisideStepFunction | title=Heaviside Step Function}}</ref> Among the possibilities are:<math display="block">\begin{align}

If we take {{math|''H''(0) {{=}} {{sfrac|1|2}}}}, equality holds in the limit:<math display="block">H(x)=\lim_{k \to \infty}\tfrac{1}{2}(1+\tanh kx)=\lim_{k \to \infty}\frac{1}{1+e^{-2kx}}.</math>There are [[Sigmoid function#Examples|many other smooth, analytic approximations]] to the step function.<ref>{{MathWorld | urlname=HeavisideStepFunction | title=Heaviside Step Function}}</ref> Among the possibilities are:<math display="block">\begin{align}

H(x) &= \lim_{k \to \infty} \left(\tfrac{1}{2} + \tfrac{1}{\pi}\arctan kx\right)\\

H(x) &= \lim_{k \to \infty}\left(\tfrac{1}{2} + \tfrac12\operatorname{erf} kx\right)