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nLab: step function

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeSep 6th 2017
- (edited Sep 6th 2017)
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Author: Urs
Format: MarkdownItexI have added to _[[step function]]_ statement and proof of its representation as $$ \begin{aligned} \Theta(x) & = \frac{1}{2\pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i 0^+} \\ & \coloneqq \underset{ \epsilon \to 0^+}{\lim} \frac{1}{2 \pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega \,, \end{aligned} $$ ([here](https://ncatlab.org/nlab/show/step+function#FourierIntegralFormula)).

I have added to step function statement and proof of its representation as
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Θ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup><mo>∫</mo> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mi>x</mi></mrow></msup></mrow><mrow><mi>ω</mi><mo>−</mo><mi>i</mi><msup><mn>0</mn> <mo>+</mo></msup></mrow></mfrac></mtd></mtr> <mtr><mtd/> <mtd><mo>≔</mo><munder><mi>lim</mi><mrow><mi>ε</mi><mo>→</mo><msup><mn>0</mn> <mo>+</mo></msup></mrow></munder><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup><mo>∫</mo> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mi>x</mi></mrow></msup></mrow><mrow><mi>ω</mi><mo>−</mo><mi>i</mi><mi>ε</mi></mrow></mfrac><mi>d</mi><mi>ω</mi><mspace width="0.16667em"/><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Theta(x) &amp; = \frac{1}{2\pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i 0^+} \\ &amp; \coloneqq \underset{ \epsilon \to 0^+}{\lim} \frac{1}{2 \pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega \,, \end{aligned} </annotation></semantics></math>$
(here).
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeSep 6th 2017
- PermaLink
Author: Urs
Format: MarkdownItexWhile I was it, I also added statement and proof that $\partial \Theta = \delta$ (copied that also over to the Properties-section of _[[delta distribution]]_ and to the Examples-section at _[[derivative of distributions]]_).

While I was it, I also added statement and proof that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∂</mo><mi>Θ</mi><mo>=</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">\partial \Theta = \delta</annotation></semantics></math>$ (copied that also over to the Properties-section of delta distribution and to the Examples-section at derivative of distributions).
- CommentRowNumber3.
- CommentAuthorDavidRoberts
- CommentTimeSep 6th 2017
- PermaLink
Author: DavidRoberts
Format: MarkdownItex[[step function]] (typo in your link)

step function (typo in your link)
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeSep 6th 2017
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Author: Urs
Format: MarkdownItexThanks, fixed now.

Thanks, fixed now.

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nLab: step function