Heaviside step function: Difference between revisions

Revision as of 14:54, 16 October 2024

Revision as of 17:07, 22 October 2024

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* using the [[Iverson bracket]] notation: <math display="block">H(x) := [x \geq 0]</math>

* an [[indicator function]]: <math display="block">H(x) := \mathbf{1}_{x \geq 0}=\mathbf 1_{\mathbb R_+}(x)</math>

* a [[hyperfunction]] <math display="block">H(x) =: \left(1-\frac{1}{2\pi i}\log z,\ -\frac{1}{2\pi i}\log z\right)</math> or equivalently <math display="block">H(x) =: \left( -\frac{\log -z}{2\pi i}, -\frac{\log -z}{2\pi i}\right)</math> where {{math|log ''z''}} is the [[Complex logarithm#Principal value|principal value of the complex logarithm]] of {{mvar|z}}

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

* a [[one-sided limit]] of the [[atan2|two-argument arctangent]] <math display="block">H(x) =: \lim_{\epsilon\to0^{+}} \frac{\mbox{atan2}(\epsilon,-x)}{\pi}</math>

For the alternative convention that {{math|''H''(0) {{=}} {{sfrac|1|2}}}}, it may be expressed as:

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

* a linear transformation of the [[sign function]],

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

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

* the [[arithmetic mean]] of two [[Iverson bracket]]s,

<math display="block">H(x) := \frac{[x\geq 0] + [x>0]}{2}</math>

Other definitions which are undefined at {{math|''H''(0)}} include:

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

* the derivative of the [[ramp function]]: <math display="block"> H(x) := \frac{xd}{dx} +\max \{ |x|, 0 \}\quad \mbox{2xfor } x \,.ne 0</math>

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 Dirac delta==