Talk:Heaviside step function Source: en.wikipedia.org/wiki/Talk:Heaviside_step_function

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The definition is self-contradictory: Who can correct it?
S.

Dirac delta function also defines it, with the ≥ switched for a ≤ - but this still doesn't match the written definition, unless I was lied to by years of math teachers and zero is negative. --Brion 08:55 Oct 26, 2002 (UTC)

Looked it up, it's =1 at zero (your teachers were right, Brion, zero, is neither neg nor pos). Funnily enough, the book I have here, they give the positive part first, then the negative, which seems the wrong way round. If I remember correctly, the value at zero isn't crucial, and different definitions exist: can be H(0)=0, H(0)=1, and H(0)=0.5 and of course, H(0)=cream cheese. needs checking though. -- Tarquin 10:16 Oct 26, 2002 (UTC)

In Japan, it is thought that

${\displaystyle \int _{-\epsilon }^{0}\delta (x)dx=\int _{0}^{+\epsilon }\delta (x)dx=1/2}$ for all ${\displaystyle \epsilon >0}$

and every image ${\displaystyle H(x)}$ of Heaviside step function ${\displaystyle H:\mathbb {R} \ni x\longrightarrow H(x)\in H(\mathbb {R} )\subset \mathbb {R} }$ is

${\displaystyle H(x)=\int _{-\infty }^{x}\delta (\xi )d\xi =\left\{{\begin{matrix}0&\left(x<0\right)\\1/2&\left(x=0\right)\\1&\left(x>0\right)\end{matrix}}\right.}$ .

Koiki Sumi 00:00 15 Sep 2003 (UTC)

I kind of doubt there's any agreement at all, even in Japan...

I absolutely agree with Japanese definition. Consider a broadened or continuous δ(x)-type function of finite width with any symmetric shape. Integrate this from minus infinity to zero, and you will see that the integral contains exactly half the δ(x) function, and so is equal to 1/2. Now take the limit as the width of the δ(x) function goes to zero, thus obtaining a true δ(x) function. The integral from minus infinity to zero remains equal to 1/2. This is exactly the value of the step function at 0. 71.32.46.85 (talk) 20:45, 5 February 2019 (UTC)Kathleen Rosser[reply]

If it can help, I've seen the "Japanese" integrals used before in pretty well respected physics books (Stochastic Methods by Gardiner, ISBN 978-3-540-70712-7, and Quantum Noise by Gardiner & Zoller, ISBN 3-540-22301-0). By the way, zero being neither negative or positive is a matter of convention. In French speaking countries (or at least France and Switzerland) 0 is considered both positive and negative :) Mojert (talk) 10:26, 13 December 2024 (UTC)[reply]

The problem with that middle is that very often in probability discrete cdfs are assumed to have one sided limit, so this would not describe discrete distributions, and afaik this convention is pretty universal (mod side) 79.191.245.209 (talk) 08:47, 31 March 2025 (UTC)[reply]

i can't find the fourier transform of the heaviside function anywhere...anyone willing to share their expertise :)

Comment at support.

Charles Matthews 13:59, 31 Jan 2004 (UTC)

Reminds me: I once won five pints of beer on a bet that it was in the textbooks (and drank three of them). As for the comment below, what is in 'the books' is typically wrong or misleading. I suppose the article can try to explain why. Charles Matthews 14:55, 11 Dec 2004 (UTC)

I was prettying up that "integral representation of the step function" at the end, and upon looking at it, i don't think it's correct. maybe it is for the Signum function , but i don't think it is for the step function. BTW, if we define the step function strictly in terms of the ${\displaystyle \operatorname {sgn}()}$, i think the Fourier Transform of it comes out nicely. also, the step function should either be undefined for x=0 or be defined to be 1/2 at x=0, but not either 1 or 0. r b-j 03:30, 11 Dec 2004 (UTC)

There appears to be an error with this section. It looks like it assumes there is some service (MathML?) running on the user's machine at port 6011:

Often an integral representation of the Heaviside step function is useful: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \begin{align} H(x)&=\lim_{ \varepsilon \to 0^+} -\frac{1}{2\pi i}\int_{-\infty}^\infty \frac{1}{\tau+i\varepsilon} e^{-i x \tau} d\tau \\ &=\lim_{ \varepsilon \to 0^+} \frac{1}{2\pi i}\int_{-\infty}^\infty \frac{1}{\tau-i\varepsilon} e^{i x \tau} d\tau. \end{align}}

where the second representation is easy to deduce from the first, given that the step function is real and thus is its own complex conjugate. — Preceding unsigned comment added by Gchoules (talk • contribs) 09:02, 30 September 2023 (UTC)[reply]

At the following address you will find the Fourier transform of the Heaviside Step Function http://mathworld.wolfram.com/HeavisideStepFunction.html

Hmmm - I'm not saying that's wrong. I would say that 1/x is not a locally integrable function. Therefore using it to represent a Schwartz distribution is not in itself a naive kind of definition. The formula therefore needs some commentary: the difference of the delta function and the reciprocal function is a combination that seems to require some discussion. Charles Matthews 12:48, 19 Apr 2005 (UTC)

All of this is weird. The Heaviside work has nothing to do with Fourier. Prof. Howie, when head of the Cavendish, told me that physical reality was composed of sine waves. http://www.ivorcatt.co.uk/774b.htm . The mathematisisation of physics has been very damaging. Fourier is about a train of identical waveforms, not about a single step. Ivor Catt 11 Aug 2016 — Preceding unsigned comment added by 109.151.217.5 (talk) 10:49, 11 August 2016 (UTC)[reply]

Why is the function letter in this article a "u"? Everywhere else i've seen an H instead.Boothinator 23:11, 26 Apr 2005 (UTC)

I was taught with a u. Probably another one of those engineer/mathematcian differences. - Omegatron 23:51, Apr 26, 2005 (UTC)

Of all the pages linking here, Dirac delta function, Distribution, Continuous Fourier transform, Sufficiency (statistics), Negative and non-negative numbers, Green's function, Sign_function (actually uses h()), Rectangular function and Uniform distribution (continuous) use the H() notation while Z-transform, Two-sided Laplace transform, User:Jacobolus/coordinates and Coordinates (elementary mathematics) use the u() notation. Recurrence plot uses a Θ(). To me, it looks like the H() notation should be used to be more consistant with the rest of Wikipedia.Boothinator 00:52, 27 Apr 2005 (UTC)

I've seen H and θ. The different notations should be mentioned and referenced. --MarSch 13:12, 30 April 2006 (UTC)[reply]

I think that H is the most commonly used notation in mathematics and θ in physics. Md2perpe 10:39, 3 August 2006 (UTC)[reply]

...and I think that ${\displaystyle u(t)}$ is the most common definition is signal processing. I, too, would support a consistent definition throughout Wikipedia. I personally like ${\displaystyle H(t)}$.--Rabbanis 20:46, 8 August 2006 (UTC)[reply]

u is more familiar to me, of course, but if a majority of articles use h I guess that's ok. I wonder if u is used to distinguish it from H(f) = Hilbert transform? — Omegatron 21:13, 8 August 2006 (UTC)[reply]

It started out as H and was changed here. I would prefer H unless you have any objections? Rex the first ^{talk | contribs} 17:14, 20 May 2006 (UTC)[reply]

I support that different notation should be mentioned. I noticed σ notation, also. --User:Vanished user 8ij3r8jwefi 16:04, 21 February 2008 (UTC)[reply]

I removed this:

There are some trials to put analytical functions to numerically calculate Unit Step Function. The study published on [1] has shown that it is possible to mimic the Unit Step Function. The results were verified using Mathematica software.

It's uninteresting, of restricted applicability, (strictly speaking) incorrect (the inverse trigonometric functions do not have unique definitions) and constitutes original research. EdC 15:14, 4 June 2006 (UTC)[reply]

It feels wrong to define H(0) as 1/2 and then immediately say the H(0) seldom matters and can be defined in various ways. However, it would also feel wrong to show a definition by case analysis which considered only x<0 and x>0. Would it be a horrible idea simply to remove the first displayed formula and just rely on the prose in the first sentence, plus the graph? Henning Makholm 01:37, 26 November 2006 (UTC)[reply]

Is it just me, or is it an incredible coinicdence that the "Heaviside" function looks like y = 1, with the negative region "heavier" (i.e. "pushed" down to 0)? The "heavier side", or "heavyside", sounds a lot like "Heaviside".... Timeroot (talk) 04:38, 27 January 2009 (UTC)[reply]

No, it's not just you. On google scholar there are 2500 "papers" using "heavyside function" and 35000 using "heaviside function".--134.58.43.57 (talk) 15:27, 26 February 2013 (UTC)[reply]

It is so painful to label a maths article as needing references,thats why i won't do it. guys, may you add more in-text references? try to get references from several sources. any one with advanced engineering text book may do better...smile :) .... Freshymail-user_talk:fngosa--the-knowledge-defender 18:16, 27 August 2009 (UTC)[reply]

Is the Heaviside step functin defined for x = 0? Looking at the alternartive definition

${\displaystyle H(x)=\lim _{z\rightarrow x^{-}}((|z|/z+1)/2)}$

H(0) would be 0. Is is sure it shouldn't be ${\displaystyle z\rightarrow x^{+}}$ or just ${\displaystyle z\rightarrow x}$? --Kri (talk) 15:30, 15 October 2009 (UTC)[reply]

Oh sorry, found the section H(0) now. Still wonder about the limit thing though. --Kri (talk) 15:34, 15 October 2009 (UTC)[reply]

Actually for the discrete case the value at H(0) does make a difference. I updated the discrete section to say that. As you said, you can fix this approximation by taking the limit from the opposite side. However this approximation doesn't seem to have anything to do with the discrete case, so I removed it from that section. I could have moved it to the "analytic approximations" section but for all x except zero, you don't need the limit. Basically I think this formula is not useful, so I've removed it entirely. Quietbritishjim (talk) 12:35, 11 February 2010 (UTC)[reply]

I removed the following assertion from the text, because I think it is false:

In particular, the measurable set

${\displaystyle \bigcup _{n=0}^{\infty }\left[2^{-2n};2^{-2n+1}\right]}$

has measure zero in the delta distribution, but its measure under each smooth approximation family becomes larger with increasing k.

If one differentiates the elements of the approximating family, then one does indeed get weak convergence to the delta measure. In fact, these all do converge as distributions as well, since their derivatives are all of the form ${\displaystyle k\eta (kx)}$ for a smooth probability distribution η. So I have also removed the following:

While these approximations converge pointwise towards the step function, the implied distributions do not converge to the Heaviside step function in the sense of distributions.

--173.75.156.204 (talk) 11:55, 22 October 2009 (UTC)[reply]

As the lead section rightly says, the value at H(0) is mostly irrelevant, and its value is usually just chosen for concreteness (if at all). I updated the H(0) section to reflect this. I toned down support for H(0)=1/2 (which seemed almost to be promotion of some author's favourite choice, and had a couple of meaningless phrases in it) and just objectively presented the reasons why each choice might be useful.

I took out the bit about using a subscript to denote the value at zero, which the article said "may be used". Well of course it may be used; a number painted in red on your forehead "may be used". Unless it's also true that it actually is used (outside of Wikipedia) its mention here is pointless. What's worse this notation is sometimes used but means something completely different: it means H(x) translated by the subscript. Quietbritishjim (talk) 01:30, 11 February 2010 (UTC)[reply]

From Heaviside_step_function#Zero_argument:

[...] it does not even make sense to talk of a value at zero, since such objects are only defined almost everywhere.

Should this not be "almost nowhere" (complement of almost everywhere)? If my point is valid, could we use the term "almost nowhere" linking this to almost everywhere? Hulten (talk) 13:42, 14 November 2016 (UTC)[reply]

> The simplest definition of the Heaviside function is as the derivative of the ramp function

There is no way that the derivative of anything is simpler than an explicit piecewise-constant definition Quietbritishjim (talk) 11:38, 21 December 2017 (UTC)[reply]

Further, the ramp function has no defined derivative at zero.--2607:FEA8:4F60:636:A93D:BB2:24A3:467A (talk) 23:15, 8 December 2019 (UTC)[reply]

What is the use of the fact: "This function is the cumulative summation of the Kronecker delta"? It seems to me to be without any sensible meaning. Madyno (talk) 08:46, 3 April 2019 (UTC)[reply]

Probability is definitely quite obvious use, pretty meaningfull as well transforming probability function of trivial RV to cdf 79.191.245.209 (talk) 08:43, 31 March 2025 (UTC)[reply]

I’ve made an equation for x≠0: ${\displaystyle {\frac {x+|x|}{2x}}}$ Meckersnapper (talk) 18:09, 14 September 2020 (UTC) Meckersnapper[reply]