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Fishsicles (talk | contribs) →Formulation: Sorting representations by H(0) values, adding a few more |
Fishsicles (talk | contribs) m →Formulation: realised the atan2 formulation was in the wrong group |
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* 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}}
* 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:
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* the [[arithmetic mean]] of two [[Iverson bracket]]s,
<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>
Other definitions which are undefined at {{math|''H''(0)}} include:
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