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Fishsicles (talk | contribs) m →Formulation: realised the atan2 formulation was in the wrong group |
Fishsicles (talk | contribs) m →Formulation: Double-checked the hyperfunction formulations against each other - they're not equivalent at H(0) |
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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
For the alternative convention that {{math|''H''(0) {{=}} {{sfrac|1|2}}}}, it may be expressed as:
* a linear transformation of the [[sign function]], <math display="block">H(x) := \frac{1}{2} \left(\mbox{sgn}\, x + 1\right)</math>
* the [[arithmetic mean]] of two [[Iverson bracket]]s, <math display="block">H(x) := \frac{
<math display="block">H(x) := \frac{[x\geq 0] + [x>0]}{2}</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>
▲* a [[hyperfunction]] <math display="block">H(x)
Other definitions which are undefined at {{math|''H''(0)}} include:
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