# Talk:Dirac delta function

Dirac delta function has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.
Article milestones
Date Process Result
September 29, 2010 Good article nominee Not listed
October 1, 2010 Good article nominee Listed
Current status: Good article

## Continuous with compact support?

Why is the the integral of a function f with respect to the Dirac measure at x not defined for all measurable f? At the moment, the text says this holds for continuous f with compact support, which seems unnecessarily restrictive. — Preceding unsigned comment added by 131.111.185.68 (talk) 09:58, 6 November 2013 (UTC)

The (admittedly somewhat unsatisfactory) answer is that this is the smallest class of functions needed to define delta as a Radon measure. But actually, since all subsets of ${\displaystyle \mathbb {R} }$ are measurable with respect to the measure, every function is measurable with respect to the measure, and the integral of f is always just f(0). (I think this might be confusing to some readers, since often in analysis functions are identified if they differ only on a set of Lebesgue measure zero, but since delta is not absolutely continuous with respect to the Lebesgue measure, such an identification can no longer be enforced. For instance, the integral with respect to delta of an f in L^1(R) is not well-defined.) Sławomir Biały (talk) 13:09, 6 November 2013 (UTC)

## Hyperfunction

I'm not following the discussion closely, so forgive me for any possible misunderstanding. But, at least to me, the article currently looks perfectly fine (organizationally speaking). The notions somehow differ from what I'm used to, but that's really not an issue. Also, I know of no mainstream definitions Dirac delta other than one by measures or distributions (maybe tempered distributions to be more precise?). I know there is the definition of Dirac in terms of Sato's hyperfunction, but this I don't think is standard. Personally, I think seeing generalized functions as limits of some nicely behaving functions is very problematic. I know Yoshida defines Sobolev spaces in this way; as the completion of some space of test functions. This doesn't work quite well in practice because there are too many choices of convergence and test functions. (Yeah, this reminded me: the article probably should relate the theory of Sobolev spaces to Dirac delta; not sure how, though. See below) I also note that the article probably spend too much spaces just for the definitions of delta, which isn't really an interesting topic after all. -- Taku (talk) 00:49, 6 July 2009 (UTC)

I didn't carefully read the article, so maybe I just missed, but there are a couple of things I think it should mention:
• Poisson summation formula in terms of delta
• How the Cauchy integral formula relates to delta
• The structure theorem for distributions, which is formulated in terms of derivatives of delta; in particular, a distribution with finite support is completely characterized by
• Delta in the representation theory: when you're dealing with group algebra, convolutions (not necessarily commutative?) involve delta.
• A better exposition of the relationship to fundamental solution; that is, why and how solving Pu = f is equivalent to solving it when f is delta
• A spherical version of the representation of delta
• ${\displaystyle \delta (x_{1},x_{2},...x_{n})=\delta (x_{1})\delta (x_{2})...\delta (x_{n})}$ (Don't know how to put this rigorously.)
• Plane-wave decomposition of the delta:
${\displaystyle \delta (x)=\int _{S^{n-1}}{\omega (\xi ) \over \langle x,\xi \rangle }}$
(where ${\displaystyle \omega }$ is a volume form of the sphere) or something like it:) It's probably due to F. John.)
-- Taku (talk) 01:25, 6 July 2009 (UTC)
I agree with most of the points you make. Actually the biggest serious omission is a proper treatment of delta in more than one variable. This problem is endemic to most of the articles here on Fourier analysis. The article Fourier transform, for instance, was clearly written with the case of one variable primarily in mind and higher dimensions only secondarily so. (I have already attempted to change this somewhat, as have other mathematics editors.)
Fundamental solutions, are already mentioned. But, since this was what emerged out of the current article structure, it is in connection with "nascent" delta functions, which is a somewhat artificial context. Perhaps the next big task will be to fork material out into a more natural section on this topic. Sławomir Biały (talk) 02:52, 6 July 2009 (UTC)
The connection to hyperfunctions, mentioned above, could be made more explicit based on the article linked at the bottom of the page. Tkuvho (talk) 15:41, 18 July 2010 (UTC)

## Laugwitz 1989?

Someone seems very keen on including a reference to Laugwitz (1989) in the "overview" section. To me, this is totally out of place. For one thing, the text

Cauchy's formula for a unit-impulse, infinitely tall, infinitely narrow delta function defined in terms of infinitesimals is reproduced by D. Laugwitz (1989), p. 230.

Doesn't fit with the surrounding paragraphs, which mentions no such formula at all. Secondly, here we are discussing Cauchy, Kirchoff, Dirac, and now Laugwitz?! The name-drop is totally jarring. I also do not think this adds anything in the way of meaningful content that is helpful for an "Overview" section. I am willing to hear reasons for the inclusion of this, but I will remove it again if I am not convinced. Please refer to WP:BRD: the editor who added this was WP:BOLD, but then reverted. Now the onus is on him or her to discuss the merits of inclusion. Sławomir Biały (talk) 12:02, 4 February 2010 (UTC)

We are not discussing Laugwitz. We are discussing Cauchy. The first paragraph in this section states that Cauchy considered sequence of taller and taller unit-impulse functions. I would like to see a source for that. In reality, Cauchy wrote down formulas for unit-impulse delta function explicitly defined in terms of infinitesimals. Note that the book you mentioned is a scientific, not a historical, monograph. The article I referred to is published in a leading historical journal. It is a more reliable source on Cauchy. Tkuvho (talk) 12:11, 4 February 2010 (UTC)
I fail to see how being a scientific rather than exclusively historical monograph disqualifies something as a source for the history of a subject. It is the only monograph (that I am aware of) that gives a detailed historical account of the Dirac delta function. It is much more detailed than the source that you are keen on citing, because it includes not only Cauchy's approach, but Poisson's, Kirchoff's, Helmholtz's, etc. Anyway, I have now included the reference you are keen on putting there in a manner that does not interrupt the flow of the text. Hopefully this compromise will be suitable. Sławomir Biały (talk) 12:23, 4 February 2010 (UTC)
You might want to consult L"utzen, J.: The prehistory of the theory of distributions. Studies in the History of Mathematics and Physical Sciences 7 Springer-Verlag, New York-Berlin, 1982. The idea that a book (van der Pol and Bremmer) written half a century ago is the last word on the subject is a little far-fetched. Unfortunately, my library does not have it. Could you tell me what they say about Cauchy exactly? As I mention below, the current version is unacceptable, as it misrepresents both Cauchy and Laugwitz. Tkuvho (talk) 12:28, 4 February 2010 (UTC)
Thank you, I will do that. Most of the van der Pol and Bremmer text is available through google books, if your library lacks it. Sławomir Biały (talk) 12:37, 4 February 2010 (UTC)
I will try to look it up. I would be very surprised if it is as detailed on Cauchy as Laugwitz's monumental 50-page text, which as I mentioned is published in a leading historical periodical. The page should contain a reference to Cauchy's infinitesimal delta function, not an ahistorical fantasy about his ideas about sequences of functions. Tkuvho (talk) 12:42, 4 February 2010 (UTC)
I see you have latched on to the word "sequences" as objectionable. Perhaps this could be changed to something more satisfactory. Sławomir Biały (talk) 12:52, 4 February 2010 (UTC)
How about mentioning the infinitesimal formula? That would have the advantage of being both more concise and more accurate. Tkuvho (talk) 13:06, 4 February 2010 (UTC)

## revert by Bialy

The following material was reverted twice by Bialy without awaiting outcome of discussion on this page:

Cauchy's formula for a unit-impulse, infinitely tall, infinitely narrow delta function defined in terms of infinitesimals is reproduced by D. Laugwitz (1989), p. 230. Tkuvho (talk) 12:17, 4 February 2010 (UTC)

The current version states that "The idea of using a sequence of functions to approximate a unit impulse goes back to the early 19th century, and was considered by Augustin Louis Cauchy", with reference to Laugwitz. This is incorrect on the counts. Laugwitz did not say Cauchy's idea was to use a sequence of functions. Instead, he reproduces a formula (not an idea) giving an infinitesimal delta function in Cauchy. Do you have a source for the current claim about Cauchy, with a citation in Cauchy? Tkuvho (talk) 12:22, 4 February 2010 (UTC)

I reverted only once. The second time, I moved the reference to against Cauchy, per your insistence in the above thread, and cut extraneous details that were irrelevant to the purpose of an "Overview" section. Sławomir Biały (talk) 12:27, 4 February 2010 (UTC)
They are not extraneous at all. On a page dealing with the dirac delta function, it is not extraneous to note that Cauchy defined it in 1827. Tkuvho (talk) 12:29, 4 February 2010 (UTC)
Interesting comment, given that no version under discussion even mentions this. Rather the version under discussion appears to give credit to Laugwitz for reproducing Cauchy's formula (which the article does not actually state) in 1989, which seems like fairly extraneous information to me. Sławomir Biały (talk) 12:40, 4 February 2010 (UTC)
I re-read the current version, and it still states that Cauchy talked about sequences of functions, with a reference to Laugwitz. Cauchy did not talk about a sequence but rather a single infinitesimal delta function, and Laugwitz was clear on this point, since he reproduces Cauchy's explicit definition in terms of infinitesimals. Tkuvho (talk) 12:45, 4 February 2010 (UTC)
Looking at the section of Laugwitz on Cauchy, the delta function appears in his derivation of the Fourier inversion formula as a limit of regularized integrals. This does seem to be consistent with the text as written. Sławomir Biały (talk) 13:05, 4 February 2010 (UTC)
Which formula are you looking at? Laugwitz mentions that Cauchy states explicitly that alpha and epsilon are infinitesimals, and writes down a single formula, not a sequence. Here epsilon is irrelevant as Laugwitz mentions later in the article. The height of Cauchy's delta function is 1/alpha. Tkuvho (talk) 13:08, 4 February 2010 (UTC)

## Cauchy before hyperreals

(moved from User talk:Sławomir Biały)

Hi, Your historical observation at Dirac delta function is perfectly correct: Cauchy came before the hyperreals. As documented at that page, Cauchy defined a Dirac-type function in terms of an infinitesimal, and a hyperreal definition thereof similarly uses infinitesimals. Thus it is inaccurate to say that a Dirac delta function cannot be defined as a true function. You don't have to be an adherent of the hyperreals to recognize an inaccuracy. Tkuvho (talk) 01:38, 9 July 2010 (UTC)

If this is an inaccuracy, it is one that is in nearly every reliable source. Anyway, I see nothing wrong with adding a separate referenced section on hyperreals, but cryptic parenthetical remarks about them in the midst of intuitive discussions are not helpful. Sławomir Biały (talk) 11:49, 9 July 2010 (UTC)
OK, thanks. Perhaps the solution is to add a brief section toward the end containing an intuitive discussion of a unit impulse, infinitely narrow, infinitely tall Dirac delta function; mention Cauchy's use of such functions in 1827; present an example in the context of a modern theory of an infinitesimal-enriched continuum; and give a bibiliography discussing infinitesimal Dirac delta fuctions starting from the 1960s. Tkuvho (talk) 13:46, 9 July 2010 (UTC)
This sounds like a good idea, except I'd recommend some caution in how Cauchy's exact conception is addressed. Cauchy explicitly stated that for him "infinitely small" meant "tending to zero". So one widely-held view (for instance, Boyer's History of the Calculus) is that Cauchy did not use literal infinitesimals (as he equally rejected "actual infinities"). Laugwitz notes both of these facets early on in his previously cited paper, but then he appears to reject this historiographical viewpoint. Anyway, it is evidently not clear that Cauchy's conception of infinitesimals squares with the modern one introduced by Robinson et al. Certainly Robinson and presumably other infinitesimal analysts have no qualms in attributing infinitesimal results to Cauchy, but this seems to rely on a literal reading of "infinitely small" that is not directly supported by Cauchy's own conception of a limit. Anyway, with appropriate care to source the history observing WP:NPOV, I see no problem in what you propose. Sławomir Biały (talk) 13:43, 10 July 2010 (UTC)
You are right, some of these issues need to be clarified before the page is modified. What, in your opinion, is Cauchy's definition of an infinitesimal? And what sources is your opinion of Robinson based upon? Tkuvho (talk) 21:29, 10 July 2010 (UTC)
In the Cours d'Analyse, Cauchy states that a variable is infinitely small if the magnitude of the variable is taken to decrease to zero. So it is not essentially different from other real-valued variables, except in the understanding that it tends to zero. This interpretation is supported by Boyer's analysis in The History of the Calculus p. 278. By opinion of Robinson is, I admit, based on a rather cursory skim of part of the previously referenced paper by Laugwitz, who in turn refers to Robinson's Nonstandard analysis text. But Robinson's text does not appear to verify Laugwitz's point, at least not straightforwardly. Rather Robinson gives what I would consider to be a fairly correct account of Cauchy's view of infinitesimals, but then explicitly reinterprets Cauchy's writings in a nonstandard setting. So I was, I think, mistaken in my original post. Sławomir Biały (talk) 14:09, 11 July 2010 (UTC)
Good, I am glad we agree on Robinson. Maybe we can try to agree on Cauchy, because Boyer's is one of many, many interpretations, differing from ones that preceded him as well as ones that followed him. So what did Cauchy say exactly? He said that a variable tending to zero "becomes" an infinitesimal. Is that a fair assessment? A recent study by K. Brating in Archive for history of exact sciences paraphrases Cauchy's definition by saying that a Cauchy infinitesimal is generated by a null sequence. Is that a fair assessment? Tkuvho (talk) 14:23, 11 July 2010 (UTC)
I think "becomes infinitely small" is the exact translation. But it's a stretch for me to read this ontologically and conclude that Cauchy thought of an infinitesimal as being some definite quantity apart from the limiting process (which "generates an infinitesimal" seems to suggest). There does appear to be consensus in the literature at least on the point that for Cauchy an infinitely small quantity was an ordinary real variable that is regarded as tending to zero. What there seems to be confusion about is the ontology: is there an "actual infinitesimal" that underlies this limiting process, or does infinitely small simply mean the manner in which the variable is treated? Lakatos' 1978 "Cauchy and the continuum" seems to have a very interesting and sophisticated take on this dilemma. He breaks down two historical trends into viewing Cauchy as either an "inarticulate Weierstrass" or an "inarticulate Robinson". I'm curious what, if any, the relevance of Lakatos' view is to our discussion. You are probably already familiar with this paper, and in a better position to answer than I. Sławomir Biały (talk) 15:26, 11 July 2010 (UTC)
I am not sure which phrase you are translating as "becomes infinitely small". The term "infinitely small" is Cauchy's exclusive term for infinitesimals. He never uses the term "infinitesimal" itself, even though it does have a French equivalent. Other authors have used it, such as L. Carnot, but not Cauchy. My reading of Cauchy is that in 1821 he defined them in terms of variable tending to zero, where the variable is understood to be a discrete variable. Thus, he gives an example which is a slight perturbation of the sequence 1, 1/2, 1/3, 1/4, etc. Note that Carnot defined infinitesimals the same way. Meanwhile, in 1823 Cauchy gave a definition in terms of a variable tending to zero, where the implied variable varies continuously. So your remark is correct with regard to the 1823 definition. Incidentally, Lakatos's 1978 text was written in the 60s (published inedited form by Cleave in 1978). Lakatos changed his mind a few times about Cauchy's infinitesimals. I personally find Laugwitz's 1989 text far more scholarly and reliable. My immediate reaction to "gut W" or "gut R" is scepticism. Why can't we take Cauchy and his definitions at face value? I am interested in the fact that Cauchy never says that a null sequence IS an infinitesimal, but only BECOMES infinitesimal. Tkuvho (talk) 16:08, 11 July 2010 (UTC)
At face value, as you say, Cauchy doesn't use the term infinitesimal, merely the term "becomes infinitely small" as a way of saying "variable tending to zero"—whether the limit is discrete or continuous doesn't seem to be especially a concern for Cauchy. Sławomir Biały (talk) 16:53, 11 July 2010 (UTC)
Again, I am not aware of Cauchy using the expression "becoming infinitely small" anywhere. Are you? Cauchy uses the expression "an infinitely small quantity" in exactly the same sense as Carnot when he uses the expression "infinitesimal", since Cauchy and Carnot give an identical definition of the respective term. Tkuvho (talk) 17:12, 11 July 2010 (UTC)
I naturally assumed that you were referring above (with the first use of "becomes" in the thread) to Cauchy's first use of the term infiniment petite in the 1821 Cours: "On dit qu'un quantite variable devient infiniment petite, lorsque sa valuer numerique decroit indefiniment de manier a converger vers la limite zero." Sławomir Biały (talk) 17:56, 11 July 2010 (UTC)
I see what you mean. What I was pointing out is that Cauchy is using "infiniment petite" here as shorthand for "quantite infiniment petite", in other words as a noun. I have the impression that most commentators agree that Cauchy is not envisioning some hitherto unmentioned "infinitesimal values" that the variable takes on its way to zero. Rather, the sequence (i.e. variable) itself is/becomes an infinitesimal. The meaning of "becoming" here is a fascinating subject, particularly since Cauchy obviously did not have access to the set-theoretic framework we take for granted today. This may not be relevant for the purposes of wiki pages, though. If we agree about the meaning of Cauchy's definition of infinitesimal, perhaps we can use it as a basis for what to write at this page. Tkuvho (talk) 18:17, 11 July 2010 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Thanks, that clarifies things for me. I think perhaps our views of Cauchy are not so radically different as I had once thought. Sławomir Biały (talk) 00:57, 12 July 2010 (UTC)

OK then, we can mention that Cauchy used an infinitesimal ${\displaystyle \alpha }$ to write down a unit impulse, infinitely tall and narrow Dirac-type delta function ${\displaystyle \delta _{\alpha }}$ satisfying ${\displaystyle \int F\delta _{\alpha }=F(0)}$ in a number of articles in 1827. Cauchy defined an infinitesimal in 1821 in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Carnot's terminology. We can mention that modern set-theoretic approaches allow one to define infinitesimals via the ultrapower construction where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a suitable ultrafilter, and perhaps mention an article or two giving a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum. Tkuvho (talk) 01:49, 12 July 2010 (UTC)
Well, let's see how it looks. Sławomir Biały (talk) 03:49, 12 July 2010 (UTC)
I was thinking of pretty much copying the material as it appears above, some like this:

### Infinitesimal delta functions

Cauchy used an infinitesimal ${\displaystyle \alpha }$ to write down a unit impulse, infinitely tall and narrow Dirac-type delta function ${\displaystyle \delta _{\alpha }}$ satisfying ${\displaystyle \int F(x)\delta _{\alpha }(x)=F(0)}$ in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Carnot's terminology.

Modern set-theoretic approaches allow one to define infinitesimals via the ultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter. The article by ... contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum such as the hyperreals. Tkuvho (talk) 04:36, 12 July 2010 (UTC)

## Two obvious problems

I saw that the article was nominated for a good article. I haven't read the entire thing, so I can't really give a proper review... but I notice two obvious problems that should be fairly easy to sort out before someone does review it properly.

• The sentence "Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ(x) that has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1." is overly long and contains too many redundant words. I know sometimes you need to explain what you have just said, but the redundancy doesn't seem to achieve that. How about "Informally, it is a generalized function that has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1."?
• The equations are numbered in an odd way. How about numbering them in numerical order, starting with 1? I don't really know how this aspect of wiki code works, but I think it would be a search and replace job to make everything right.

Yaris678 (talk) 17:37, 11 July 2010 (UTC)

I have changed the lead sentence in line with my first point. Yaris678 (talk) 11:11, 13 July 2010 (UTC)
I notice that Sławomir Biały has fixed the equation numbering with this change. Thanks. Yaris678 (talk) 11:56, 13 October 2010 (UTC)

## Representations of the delta function

This section seems a bit weak to me. It should be connected with expressions involving sums of eigenfunctions, for example:

${\displaystyle \delta (x-\xi )=2\sum _{1}^{\infty }\sin n\pi x\ \sin n\pi \xi \ .}$
${\displaystyle \delta (x-\xi )={\frac {2}{\pi }}\int _{0}^{\infty }\ \sin kx\sin k\xi \ dk\ .}$

These are generated from the completeness relations for various sets of functions, for example eigenfunctions, and also are closely related to Green's functions:

${\displaystyle \delta (x-\xi )=-{\frac {1}{2\pi }}\oint G_{\lambda }(x,\ \xi )\ d\lambda \ .}$ Brews ohare (talk) 15:44, 12 July 2010 (UTC)

For an inadequate discussion see Resolution of identity and this. A very particular example:

${\displaystyle \delta (x-x')=\oint _{C}\psi _{+}(x,\ k)\psi _{-}(x',\ k)\ dk\ ,}$

is found here. A basic intro is here and here. Brews ohare (talk) 16:13, 12 July 2010 (UTC)

I added a section on eigenfunction representations, but have not dealt with Green's functions. Brews ohare (talk) 15:08, 13 July 2010 (UTC)

Nice. I've tried to make it a little less formal. Green's functions and fundamental solutions should be treated together, and I agree this is a major omission. However, for what it's worth, I think an explicitly "spectral" approach to them may be unwarranted. There already is some scattered discussion of fundamental solutions of evolution equations, but there needs to be some discussion to tie this together. Sławomir Biały (talk) 15:50, 13 July 2010 (UTC)
You've tightened this up nicely. Can you add some explanation or sources or links to guide the reader to "converges in the distributional sense"? Brews ohare (talk) 16:08, 13 July 2010 (UTC)

## Identity removed

I have reorganized the material to what seems to be a more logical order (e.g., discussing the distribution derivative of the delta function in the section that treats derivatives). In the process, I removed the identity

${\displaystyle {\frac {d}{dx}}\log x={\frac {1}{x}}+i\pi \delta (x).}$

As stated, it doesn't make sense because 1/x is ambiguous. I had initially changed this to a principal value, but I think that's wrong. Rather it seems to be getting at Sokhostky's formula, which I will add a short section on. Sławomir Biały (talk) 16:33, 13 July 2010 (UTC)

## McMahon reference

Reference 29 to McMahon appears to be misplaced as it is simply a general intro to the delta function and doesn't refer to the rectangular approximation nor to statistical usage. Brews ohare (talk) 13:28, 14 July 2010 (UTC)

I replaced this reference with this one. Brews ohare (talk) 14:46, 14 July 2010 (UTC)

## Inner products with a weight function

Here is a possible re-write of a portion of the "Resolutions of the identity" section:

...the expression for f can be rewritten as:

${\displaystyle f(x)=\int _{D}\,d\xi \ w(\xi )\ \left(\sum _{n=1}^{\infty }\ \varphi _{n}(x)\varphi _{n}^{*}(\xi )\right)f(\xi ).}$

where provision is made for a weight function w(x) in the inner product. The right-hand side converges to f in the L2 sense. It need not hold in a pointwise sense, even when f is a continuous function. Multiplying both sides by the weight function:

${\displaystyle w(x)f(x)=\int _{D}\,d\xi \ w(\xi )f(\xi )\ \left(w(x)\sum _{n=1}^{\infty }\ \varphi _{n}(x)\varphi _{n}^{*}(\xi )\right).}$

It is common to abuse notation and write

${\displaystyle w(x)f(x)=\int \ d\xi \ w(\xi )f(\xi )\ \delta (x-\xi ),}$

resulting in the representation of the delta function:

${\displaystyle \delta (x-\xi )=w(x)\sum _{n=1}^{\infty }\ \varphi _{n}(x)\varphi _{n}^{*}(\xi ).}$

Although this looks superficially OK, the symmetry in x and ξ is lost, with the result that integral of the delta function over x is 1, but the integral over ξ is not 1.

Can this be fixed up and inclusion of a weight function made part of the article? Brews ohare (talk) 16:28, 14 July 2010 (UTC)

Should the first formula be
${\displaystyle w(x)f(x)=\sum _{n=1}^{\infty }\int _{D}\,d\xi \ w(\xi )\ \left(\ \varphi _{n}(x)\varphi _{n}^{*}(\xi )\right)f(\xi )?}$
By the way I think the sum should go outside the integral- it doesn't converge except in a weak sense. Then the last formula would be
${\displaystyle \delta (x-\xi )={\frac {w(\xi )}{w(x)}}\sum _{n=1}^{\infty }\ \varphi _{n}(x)\varphi _{n}^{*}(\xi ).}$
Seems to me that the initial formula you wrote would only work if ${\displaystyle \varphi _{n}}$ was an o.n. basis with respect to the weighted inner product. Do you have a source? Holmansf (talk) 17:11, 14 July 2010 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────Holmansf: The source Davis & Thompson is typical. As you point out, the discussion is limited to the case:

${\displaystyle \int _{D}\ d\xi \ w(\xi )\ \phi _{n}(\xi )\phi _{m}^{*}(\xi )=\delta _{nm}\ ,}$

with {${\displaystyle \phi _{n}}$} a complete set. That established,

${\displaystyle f(x)=\sum _{n=1}^{\infty }\ \phi _{n}(x)\int _{D}\ d\xi w(\xi )\phi _{n}^{*}(\xi )f(\xi )\ .}$

Thus, one finds:

${\displaystyle w(x)f(x)=\sum _{n=1}^{\infty }\ w(x)\ \phi _{n}(x)\int _{D}\ d\xi w(\xi )\phi _{n}^{*}(\xi )f(\xi )\ .}$

I'd guess that in most cases of interest the sum and the integration can be interchanged. I do not know the exact technical requirements for that to be OK, but it is done all the time in physics texts without comment. Assuming some cases exist where interchange is OK, one finds:

${\displaystyle h(x)=\int _{D}\ d\xi \ \sum _{n=1}^{\infty }\ \ w(x)\phi _{n}(x)\phi _{n}^{*}(\xi )h(\xi )\ ,}$

with h(x) = w(x)f(x), suggesting:

${\displaystyle \delta (x-\xi )=\sum _{n=1}^{\infty }\ \ w(x)\phi _{n}(x)\phi _{n}^{*}(\xi )\ .}$

The source does not make this identification as they treat the case where w(x) = 1 in talking about the delta function, and do not use the delta function when w(x) ≠ 1. As you can see, this result cannot be true because of the asymmetry in the two arguments, as also is true of your suggested alternative.

Maybe the correct approach is to say the delta function is like any other function so that:

${\displaystyle (f,\delta (x-x_{0}))=\int _{D}\ d\xi \ w(\xi )\ f(\xi )\delta (\xi -x_{0})\ .}$

With the δ-function:

${\displaystyle \delta (\xi -x_{0})=\sum _{n=1}^{\infty }\ \ \phi _{n}(x_{0})\phi _{n}^{*}(\xi )\ ,}$

this produces:

${\displaystyle (f,\delta (x-x_{0}))=\sum _{n=1}^{\infty }\ \ \phi _{n}(x_{0})c_{n}=f(x_{0})\ ,}$

as required.

Any suggestions? Brews ohare (talk) 19:44, 14 July 2010 (UTC)

Here is a sample re-write of this section. Please comment. Brews ohare (talk) 20:16, 14 July 2010 (UTC)

The formula I wrote is correct despite your reservations about asymmetry. It's actually easy to see it's correct since if you take any smooth function ${\displaystyle F(x,\xi )}$ such that ${\displaystyle F(x,x)=1}$ for all x, then ${\displaystyle F\delta =\delta }$. ${\displaystyle F(x,\xi )=w(\xi )/w(x)}$ is such a function.
Some of the formulas you have in your rewrite are not quite correct. Also, you should use some notation like ${\displaystyle L_{w}^{2}(D)}$ to indicate you are in a weighted space. Personally, I'd leave the section as is, and then add a comment about weighted spaces at the end.
Also, I'd make the example much shorter. You could put in the final formula giving the delta as a sum, but I think the details you have should go in Fourier-Bessel series.
The sum should go outside the integral. Otherwise it's just a formal notation and not an actual integral. Holmansf (talk) 21:36, 14 July 2010 (UTC)

I lean in the direction of doing all this using the Dirac measure, although I am as yet unclear just how to travel that path. This may help as may this. Brews ohare (talk) 01:15, 15 July 2010 (UTC)

I always think of the delta function as a distribution, and am not really sure of the advantage of regarding it as a measure. I am moving the sum outside the integral in accordance with my comments above. Holmansf (talk) 02:16, 15 July 2010 (UTC)

## GA Review

Reviewer: tb240904 (talk) 01:53, 26 July 2010 (UTC)

Criteria taken from Wikipedia:Good_article_criteria

### 1. Well-written

(a) the prose is clear and the spelling and grammar are correct

no problems found

(b) it complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation

Overall:

### 2. Factually accurate and verifiable

(a) it provides references to all sources of information in the section(s) dedicated to the attribution of these sources according to the guide to layout

(b) it provides in-line citations from reliable sources for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons-science-based articles should follow the scientific citation guidelines

Could do with some more inline citations

(c) it contains no original research

Overall:

### 3. Broad in its coverage

(a) it addresses the main aspects of the topic

(b) it stays focused on the topic without going into unnecessary detail (see summary style)

Goes into detail but I can't see anything that's obviously unnecessary.

Overall:

### 4. Neutral

It represents viewpoints fairly and without bias.

no problems

Overall:

### 5. Stable

It does not change significantly from day to day because of an ongoing edit war or content dispute.

several editors working on this with no edit warring

Overall:

### 6. Illustrated, if possible, by images

(a) images are tagged with their copyright status, and valid fair use rationales are provided for non-free content

File:Dirac_distribution_PDF.svg - CC Attribution Share Alike
File:Dirac_function_approximation.gif - Public domain, released by author

(b) images are relevant to the topic, and have suitable captions

File:Dirac_distribution_PDF.svg - relevant, with caption.
File:Dirac_function_approximation.gif - relevant, with caption.

Overall:

### 7. Conclusion

This article may not meet the following points of the good article criteria:

• 2(b)

Overall: Article is on hold Overall: Article failed 01 October 2010

--tb240904 Talk Contribs 17:23, 1 October 2010 (UTC)

### 8. Notes for the Editor

• More inline citations
• Separate notes and citations
This appears to have been already done. Sławomir Biały (talk) 20:19, 24 August 2010 (UTC)
The notes section expands on the information given in the article (i.e. notes) and gives references for individual material (i.e. references). There is then a reference section which gives details on the books referenced. --tb240904 Talk Contribs 17:32, 1 October 2010 (UTC)

Reviewer: tb240904 Talk Contribs 01:53, 26 July 2010 (UTC)

What's the status on the review? No updates on this page in nearly a month. Wizardman Operation Big Bear 04:57, 18 August 2010 (UTC)
I have been out of town, and apparently the only one around with an interest in getting this GA to go through. Give me a week or so, and I will bring it into shape. Sławomir Biały (talk) 21:07, 24 August 2010 (UTC)
This review has been going on for far too long. All editors involved should make every effort to close this review as soon as possible. Also, keep in mind that we have special citation guidelines for math-related articles. Edge3 (talk) 15:14, 9 September 2010 (UTC)
Agreed, it should be closed one way or the other within the next few days. Wizardman Operation Big Bear 17:53, 13 September 2010 (UTC)
I have left another note on the reviewer's talk page. Jezhotwells (talk) 19:48, 27 September 2010 (UTC)
OK, as the reviewer is not responding and as I personally don't feel capable of reviewing this. I am going to close it as a failed nomination now. Jezhotwells (talk) 01:59, 29 September 2010 (UTC)
I have renominated, with a timestamp one hour later to allow this keep its place in the queue. Jezhotwells (talk) 02:27, 29 September 2010 (UTC)
I appologise for not responding but my home computer is currently unable to connect to the internet and I was unable to access my user talk page due to my school's firewall. I only had concerns about the references/notes sections but I see the article has been reviewed by another editor and passed. --tb240904 Talk Contribs 17:32, 1 October 2010 (UTC)

## Application to quantum mechanics

Application to quantum mechanics

==Application to quantum mechanics==

We give an example of how the delta function is expedient in quantum mechanics. A set {φn} of orthonormal wave functions is complete in the space L2 of square-integrable functions if any wave function ψ can be expressed as a combination of the φn:

${\displaystyle \psi =\sum c_{n}\phi _{n},}$

with ${\displaystyle c_{n}=\langle \phi _{n}|\psi \rangle }$. Complete orthonormal systems of wave functions appear naturally as the eigenfunctions of the Hamiltonian (for bound states) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the spectrum of the Hamiltonian. In bra-ket notation, as above, this equality implies that

${\displaystyle \delta =\sum |\phi _{n}\rangle \langle \phi _{n}|.}$

For other observables in quantum mechanics, the set of eigenfunctions may be continuous rather than discrete. An example is the position observable, (x) = xψ(x). The spectrum of the position observable (in one dimension) is the entire real line, and is called a continuous spectrum. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well: that is, to replace the Hilbert space of quantum mechanics by an appropriate rigged Hilbert space.[1] In this context, the position operator has a complete set of eigen-distributions, labeled by the points y of the real line, given by

${\displaystyle \phi _{y}(x)=\delta (x-y).}$

The eigenfunctions of position are denoted by ${\displaystyle \phi _{y}=|y\rangle }$ in Dirac notation, and are known as position eigenstates. Although they are not physical particle states, they are fundamental in quantum mechanics.

Similar considerations apply to the eigenstates of the momentum operator, or indeed any other observable P with a complete set of eigen-distributions. What this means is that there is a set {φy} of functions, labeled by y in some subset Ω of the complex plane such that

${\displaystyle P\phi _{y}=y\phi _{y}}$

and, for any test function ψ,

${\displaystyle \psi (x)=\int _{\Omega }c(y)\phi _{y}(x)\,dy}$

where

${\displaystyle c(y)=\langle \psi ,\phi _{y}\rangle }$.

If the eigenvectors are normalized so that

${\displaystyle \langle \phi _{y},\phi _{y'}\rangle =\delta (y-y')}$

in some sense, then there is a resolution of the identity

${\displaystyle \delta =\int _{\Omega }|\phi _{y}\rangle \,\langle \phi _{y}|\,dy}$

where the operator-valued integral is understood in the weak sense.

The delta function also has many more specialized applications in quantum mechanics, such as the delta potential models for a single and double potential well.

I moved to here the section Application to quantum mechanics because it has so many problems and is misinformation. For example, in the following equations from the section

${\displaystyle \delta =\sum |\phi _{n}\rangle \langle \phi _{n}|}$

and

${\displaystyle \delta =\int _{\Omega }|\phi _{y}\rangle \,\langle \phi _{y}|\,dy}$

the left hand sides should be 1, not ${\displaystyle \delta }$. And if these equations are corrected they lose their purpose in the context.

In my opinion, it would be best to completely rewrite the section. --Bob K31416 (talk) 19:16, 3 September 2010 (UTC) P.S. Instead of generating error-filled OR, I would suggest using sources. For example, Dirac, Paul A. M. (1967). Principles of Quantum Mechanics (revised 4th ed.). London: Oxford University Press. p. 58. ISBN 978-0198520115. or college textbooks on quantum mechanics. --Bob K31416 (talk) 20:46, 3 September 2010 (UTC)

I guess I assume some responsibility for some of the errors in the section. This was my own effort to rewrite what was here before in a way that actually made sense, although I do apologize for the errors that I may have introduced in my carelessness along the way. I support the idea of rewriting from sources, but the emphasis that most sources have often makes it very difficult to write material on the delta function itself, as it is seldom the focus. It seems to me the basic assumption in quantum mechanics is that it is possible to normalize the eigenstates of a self adjoint operator (with a continuous spectrum) so that
${\displaystyle \langle x|y\rangle =\delta (x-y).}$
One can then resolve any state as a superposition of these eigenstates by the resolution of the identity. This is, at least, what it seems to me that the earlier revision had intended to convey, and it was this sense that I had intended to preserve. My own carelessness in the exposition notwithstanding, this particular conclusion is surely not "original research". Whether this bears enough relevance to this particular article is a valid concern, though. Sławomir Biały (talk) 22:39, 3 September 2010 (UTC)
I've attempted again to improve the section, hopefully with fewer errors this time. Constructive feedback, as always, is most appreciated. Sławomir Biały (talk) 01:17, 4 September 2010 (UTC)
Why did you use an undefined quantity "I" instead of 1? What is the purpose of the two corresponding equations in discussing the delta function? --Bob K31416 (talk) 03:52, 4 September 2010 (UTC)
I is fairly standard notation for the identity operator, and we already use this in the section on Resolutions of the identity. The notation "1" might be taken to refer to the function f(x) = 1 in the rigged Hilbert space. Sławomir Biały (talk) 10:52, 4 September 2010 (UTC)
And your response to my other question re purpose is....? --Bob K31416 (talk) 13:40, 4 September 2010 (UTC)
The description already in the Resolutions of the identity section seems apropos. The operator identity
${\displaystyle I=\int |x\rangle \langle x|\,dx}$
is equivalent to the normalization
${\displaystyle \langle x|y\rangle =\delta (x-y)}$
by applying each equality to a pair of test functions. Sławomir Biały (talk) 14:18, 4 September 2010 (UTC)

<outdent>That doesn't appear to be quite true. If you feel they are equivalent, try deriving each from the other. However, from looking at the "Resolutions of the identity" subsection of the Hilbert space theory section and the source there " Davis & Thomson 2000, Equation 8.9.11, p. 344" , I think I see what the original forms of the equations at the top of this Talk section were trying to do.

${\displaystyle \delta (x-\xi )=\sum _{n=1}^{\infty }\ \varphi _{n}(x)\varphi _{n}^{*}(\xi ).}$

I'm not sure of the similar expression for the continuous index case, which isn't mentioned in the "Resolutions of the identity" subsection. Is there a source for the continuous index case? --Bob K31416 (talk) 16:12, 4 September 2010 (UTC)

Sorry, my mistake. The eigenvectors need to be suitably complete in order to show equivalence: every compactly supported function on the spectrum must have the form ${\displaystyle \phi (x)=\langle \phi |x\rangle }$ for some ${\displaystyle |\phi \rangle }$. I think this is true automatically for bounded self-adjoint operators by the Gelfand-Naimark theorem. I don't know about the unbounded case, which is the primary case of importance here. Sławomir Biały (talk) 21:50, 4 September 2010 (UTC)
I'm not sure how to put my feelings about the situation at this article. I think I'll just leave. I do recognize that you are trying but I think there is a fundamental problem here. Anyhow, good luck. --Bob K31416 (talk) 00:59, 5 September 2010 (UTC)

Some remarks:

• Quantum mechanics does not deal exclusively with the Hilbert space L2. Spin systems often have finite dimensional Hilbert spaces. For this reason it is not immediately clear how the given example is situated in physics. Some attempt to setup up the physical context (e.g. a free particle) could be made.
• The Hamiltonian itself may have a continuous spectrum. (For example, the Hamiltonian of a free particle has a continuous spectrum) This makes the sentence: "For other observables in quantum mechanics, the set of eigenfunctions may be continuous rather than discrete." a bit weird.
• Strictly speaking, neither position nor momentum operators are self-adjoint, because their adjoint is not well-defined. Within this context a looser definition of self-adjoint applies.
• This appearance of the delta function in QM is hardly the most notable (although it is important.) At least equally important is the appearance of the Dirac delta in the canonical commutation relations, and the role of the Dirac delta in the definition of Green's functions. (The latter is actually an application of the Delta function that is important in nearly any field of physics.
• It would greatly help if the section were based of a RS. Using Dirac's book might not be the best idea though.TimothyRias (talk) 11:40, 6 September 2010 (UTC)

Thanks for the remarks. Three of these issues are fairly easily addressed. I have attempted to set up the physical context for your first remark. Also, in the second remark, An earlier revision said that this was the Hamiltonian of a bound state, but that somehow didn't make it into the version you saw: I have restored it. For the third bullet, there is a standard way of treating self-adjoint unbounded operators (provided they are densely defined). Usually (at least in my experience) "self-adjoint" means that the operator can be unbounded, whereas "hermitian" implies boundedness (or do I have it backwards?) At any rate, I have added an appropriate link to the unbounded operator article.
I have a question regarding your fourth point. Presumably you mean the canonical commutation relations of field theory (i.e., "second quantization")? Sławomir Biały (talk) 13:20, 6 September 2010 (UTC)
Tweaks look good. The phrase "Hamiltonian of a bound state" is a bit peculiar. I guess it is supposed to convey the idea that the eigenstates of the Hamiltonian are bound states. Wouldn't "Hamiltonian of a bound system" be better? As for your last question, yes I was referring to the canonical commutation relations of a quantum field theory, as used in canonical quantization. (I really hate the term "second quantization" since it is somewhat misleading.) As for the standard usage of either the term self-adjoint of Hermitian for a densely defined self-adjoint operator: Don't know, physicists generally use the term Hermetian in this context (and simply ignore any subtleties relating to this issue).TimothyRias (talk) 13:38, 6 September 2010 (UTC)

I agree with some of the edits to the lead, but disagree with others. For one thing, the lead sets up a contrast between the informal description of the delta function as a quantity that is zero at every nonzero point, but with total integral one, and its actual definition as a measure or distribution. The article discusses this in great detail. I have added a reference to this effect. It is important to note that Dirac's 1927 textbook is not the ultimate authority on the delta function: much has been done in both mathematics and physics to lay this on a more satisfactory foundation. Sławomir Biały (talk) 11:06, 4 September 2010 (UTC)

I think the 1st edition of Dirac's book was in 1930 and the last edition that Dirac revised was in 1967.
Here's an excerpt from Dirac's discussion of the delta function on p.58 in the 1967 edition of his book.
"δ(x) is not a function of x according to the usual mathematical definition of a function, which requires a function to have a definite value for each point in its domain..."
The corresponding sentence of mine that you removed[1] was
"The δ function is not a mathematical function according to the usual definition because it doesn't have a definite value when its parameter is zero."
In your edit summary you wrote, "I don't agree with this edit. Lack of a definite value is not essential ...".
It appears that you disagree with Dirac's comment that the definition of a function requires that a function have a definite value for each point in its domain. Could you give a source and the excerpt from the source that contains the idea, that you seem to have, that the definition of a function does not require a function to have a a definite value for each point in its domain? Thanks. --Bob K31416 (talk) 14:28, 4 September 2010 (UTC)
I'm not disagreeing that functions (as conventionally defined in most areas of mathematics) have values at each point of the domain. What I'm disagreeing with is that this is at all relevant to the discussion. Elements of the space L1, for instance, while commonly called "integrable functions", are actually equivalence classes of functions that are mutually equal almost everywhere. Thus such "functions" indeed lack a definite value at any given point of the domain. Such functions have a well-defined integral, and in fact have the aforementioned property: that if they are zero at every point save one, then they must have total integral zero. (An example of an L1 function from physics is the Heaviside function on the interval [−1,1]. In practice, it seldom matters how (or even if) the Heaviside function is defined at the origin.) However, whereas these L1 functions fit neatly into conventional integration theory, the delta function actually requires a different formalism altogether to put on a proper foundation. The latter point is more substantial from the point of view of measure theory and distribution theory. So, while Dirac's observation that the delta function is not a function because it lacks a specific value at one point of the domain may be technically correct, it emphasizes the wrong aspect of the theory from the point of view of the modern parallel developments of integration theory and the theory of distributions. Sławomir Biały (talk) 15:53, 4 September 2010 (UTC)
It's not clear what your position is regarding the definition of a function. You began by saying, "I'm not disagreeing that functions (as conventionally defined in most areas of mathematics) have values at each point of the domain." You continue with remarks that don't seem to be consistent with this beginning statement. With the phrase "(as conventionally defined in most areas of mathematics)" perhaps you feel that there is a definition of function that does not require a function to have a definite value for each point in its domain?
As you noted there are mathematical objects that have the word function in their names that don't have a definite value for each point in their respective domains, but please note that like the delta function, this does not mean they are functions according to the usual definition. My edit made it clear at the beginning, and in a way that coordinated well with the previous sentence, that the delta function is not a function according to the usual definition, and was simply reporting what the source correctly stated. The next paragraph of the lead in that previous version of the article that was reverted, was there before my edit and satisfies the other subjects that you brought up. Regards, --Bob K31416 (talk) 17:30, 4 September 2010 (UTC)
The fact that the delta function is of "indeterminate value" at a particular point misses the mark. The integral of a function whose value at a single point is unspecified (or even is infinite) and which is zero everywhere else is well-defined, and the result is zero. Strictly speaking, it doesn't even make sense to talk about the "value" of the delta function at any point: it is a distribution (mathematics) pure and simple, not a pointwise object at all. Compare this situation with actual functions and actual integrals: simply lacking a value at a point does not turn a function into a more mysterious object, it simply changes the domain of the function (and not its integral). Sławomir Biały (talk) 20:10, 4 September 2010 (UTC)
It doesn't look like we're getting anywhere in this discussion. --Bob K31416 (talk) 20:27, 4 September 2010 (UTC)
To put it another way, the delta function isn't just "not a function", it isn't even a locally integrable function. Sławomir Biały (talk) 20:45, 4 September 2010 (UTC)
That brings to mind an issue regarding the definition of the delta function. Its defined in terms of an integral that does not exist, according to the definition of an integral, since the integrand does not have a definite value at x=0. Perhaps that was your point all along? --Bob K31416 (talk) 21:49, 4 September 2010 (UTC)
Well yes. Something like: if the integral existed, then it would necessarily be zero by the definition of the integral (e.g., Lebesgue integral). Sławomir Biały (talk) 21:52, 4 September 2010 (UTC)
I was thinking of the definition of the Riemann integral with regard to it not existing, rather than being zero. --Bob K31416 (talk) 22:22, 4 September 2010 (UTC)
P.S. I don't readily have available the Vladimirov reference that appears in the 2nd paragraph of the lead. Could you give the excerpt from that reference that supports the point that the integral is zero? --Bob K31416 (talk) 22:38, 4 September 2010 (UTC)

After motivating the definition of the Delta function as a limit of step functions, Vladimirov on page 64 says:

We shall first take the density to be point limit of the sequence of average densities fε, that is,

${\displaystyle \delta (x)=\lim _{\epsilon \to 0}f_{\epsilon }(x)={\begin{cases}+\infty &\mathrm {if} \ x=0\\0&\mathrm {if} \ x\not =0\end{cases}}}$

(1)

It is naturally required that the integral of the density δ over any volume G should give the mass of this volume, that is,
${\displaystyle \int _{G}\delta (x)\,dx={\begin{cases}1&\mathrm {if} \ 0\in G\\0&\mathrm {if} \ 0\not \in G\end{cases}}}$
But, by virtue of (1), the left-hand side of this equation is always equal to zero if the integral is taken to be improper.

Vladimirov then proceeds to define generalized functions, and distinguishes between regular and singular generalized functions: the regular ones being those that come from locally integrable functions. Ultimately, on page 74, he presents a more careful argument that the delta function is not regular which proceeds by showing under the assumption that δ is regular, that is equal to zero almost everywhere, but that this contradicts the definition of the delta function. Sławomir Biały (talk) 12:58, 6 September 2010 (UTC)

## Disambiguation really needed

I have noticed that a hatnote directing readers to the Kronecker delta has been added twice to this article. I am inclined to removed this, per WP:RELATED. The article already amply covers the Kronecker delta in a section. Is there any real concern that someone will have arrived here in error? Sławomir Biały (talk) 13:10, 11 October 2010 (UTC)

There's a huge problem in Wikipedia with technical articles being too technical and confusing for non-expert users. Check out WP:TECHNICAL and Wikipedia:Lead section#Introductory text and you'll see some info on how to make technical articles better. In technical articles it's vitally important that the intro cover the most pertinent issues and be directed towards the non-expert, who makes up the vast majority of the readers of the article. This includes what a topic is (explained, if at all possible, in a way that makes sense to the non-expert), why a topic is important, what it's useful for, and any other considerations relevant to the non-expert. In this case it's very common for non-expert readers to confuse the Dirac delta and Kronecker delta, and thus this needs to go in the lead. It doesn't have to be a hatnote -- imo even better might be a short section that actually explains the difference. In fact, I just changed the article correspondingly.
Now you might object that we already have a section comparing the two. The problem is (1) this section is way down towards the end rather than near the beginning; and (2) it's phrased in a way that will make no sense to a non-expert user. Furthermore, there's nothing wrong with having the comparison mentioned twice. In fact, it's probably exactly the right thing. A Wikipedia article should follow the inverted pyramid structure of a newspaper article, with the most important info first, summarized in the lead and then explained in more detail later on. The lead should stand on its own -- see WP:LEAD. Benwing (talk) 04:06, 12 October 2010 (UTC)
The tone of your reply suggests that I am making an argument that I never did. Nowhere in WP:MTAA or WP:LEAD does it say that we should populate our technical articles with irrelevant hatnotes. I have no problem mentioning the Kronecker delta in the lead, although an entire paragraph on the subject is disproportionate to the coverage in the article (and even level of interest and importance of the Kronecker delta vis-a-vis the Dirac delta). So I've trimmed this down somewhat and put it in what seems to be a natural place.
Regarding making technical articles more accessible, I think we should focus there on the big picture. The Kronecker delta is of strictly peripheral importance: even if it is more "accessible", moving that content upwards without a view for how that enhances an understanding of the Delta function is not likely to lead to improvement. Instead someone needs to take a stab at improving the introductory section, with a care to sourcing it properly and including only content that is part of the "standard" treatment of the subject. Some pseudo-mathematical examples might be helpful as well, with illustrations showing limits of Gaussians or hat functions tending to the delta function. If I recall, the Strichartz text referenced in the article has a fairly good introductory account, and might be useful as a source for writing such a section. Then a good source for the history needs to be found. At one point, I had suggested van der Pol and Bremmer, which has a rather good account of the early history. I'm told that Lutzen's "Prehistory of the theory of distributions" might be better. Probably any textbook on distributions will give some account of the more recent history. Thirdly, someone needs to focus on writing an Applications section, including the current application to quantum mechanics, and the stublike application to probability, as well as including applications to other diverse areas like Green's functions and partial differential equations. Once all this has been done, hopefully a natural article structure will emerge that will facilitate greater accessibility. Sławomir Biały (talk) 12:03, 12 October 2010 (UTC)
I agree: Kronecker delta is peripheral here. Tkuvho (talk) 13:39, 12 October 2010 (UTC)

## the integral of the Kronecker delta is equal to zero?

This phrase from the introduction is misleading. Usually Kronecker delta is useful in the case of a finite domain, where one should be "integrating" with respect to point measures. Tkuvho (talk) 13:33, 12 October 2010 (UTC)

I agree that one should not think of the Kronecker delta on a continuous domain. The fact that the integral of such a function over a continuous domain is zero emphasizes the dissimilarity of the Kronecker delta and Dirac delta rather than their fundamental similarity: in some sense the Dirac delta is the proper generalization of the Kronecker delta from a discrete domain to a continuous one. Sławomir Biały (talk) 11:27, 13 October 2010 (UTC)

## Informally?

This article starts: "The Dirac delta function, or δ function, is (informally)..." At this point, one would hope most readers would either roll their eyes, start laughing, or just stop reading. Wiki is at best "informal" to begin with - this isn't a mathematics journal or textbook. If this definition is informal in an already informal environment - what is it? Well, being wiki I could tell you what it is, but some children might read this in the future - let's just say it rythms with nit and wit. —Preceding unsigned comment added by 192.158.61.139 (talk) 20:48, 13 January 2011 (UTC)

What could rhyme with nit and wit that could possibly describe a definition in such an offensive way? genuinely curious here. — Preceding unsigned comment added by 60.184.245.143 (talk) 15:23, 25 January 2012 (UTC)

Bullshit. Bill Wvbailey (talk) 23:14, 25 January 2012 (UTC)

## dirac delta function == sum of exponentials

Could someone please check this formula ?

${\displaystyle \delta (x)={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{inx}.}$

I would suppose the RHS gives rise to a whole comb of delta-functions: Setting x=2*pi definitely does not give zero. Benjamin.friedrich (talk) 12:50, 7 February 2011 (UTC)

This gives the delta function on the circle rather than the real line (ie x is in [-pi,pi]). If x is regarded as a variable on the real line, then this is the periodization of the Dirac delta, which is is the Dirac comb. The Poisson summation formula is relevant as well. Sławomir Biały (talk) 13:28, 7 February 2011 (UTC)
I have added a statement that hopefully clarifies this assertion in the text. It is clear from the context that it intends to refer to functions on the circle. Sławomir Biały (talk) 13:58, 7 February 2011 (UTC)

## Composition with a function example slightly confusing

Hi all, excuse me if I make any blunders - this is my first attempt at creating a discussion section.

We are told the change of variables shown holds " provided that g is a continuously differentiable function with g′ nowhere zero. " However the subsequent example sets ${\displaystyle g(x)=x^{2}-\alpha ^{2}}$ which does not obey the second requirement that the first differential is nowhere zero.

This is remedied by the fact that the requirement for change of variables need only be true over the domain of integration, not necessarily the whole real line as shown in the equation

${\displaystyle \int _{\mathbf {R} }\delta {\bigl (}g(x){\bigr )}f{\bigl (}g(x){\bigr )}|g'(x)|\,dx=\int _{g(\mathbf {R} )}\delta (u)f(u)\,du}$

In the example, the function ${\displaystyle g(x)}$ need only satisfy the above requirements in the neighbourhood of α and -α. I suggest replacing ${\displaystyle \mathbf {R} }$ with an arbitrary domain such as ${\displaystyle \mathbf {\Omega } }$ in order to clarify this, but please correct me if you believe I'm wrong.--Gabs (talk) 15:17, 9 June 2011 (UTC)

The definition does not require that the derivative of g be nonzero, only that the roots be simple. This is only used to motivate writing down the definition. Sławomir Biały (talk) 15:25, 9 June 2011 (UTC)
I'm not sure I understand you. The change of variables method requires that the derivative is nowhere zero on the domain of integration - this is what the article states and references. In practise, if the derivative is zero at a finite number of points then we split the domain of integration around these points.Gabriel Rosser (talk) 10:09, 10 June 2011 (UTC)
From the article,
It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions g by
${\displaystyle \delta (g(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|g'(x_{i})|}}}$
where the sum extends over all roots of g(x), which are assumed to be simple.
This is referenced to Gelfand and Shilov, who also define the composition in this way. This definition doesn't involve change of variables in an integral. That was just to motivate writing this definition down. Sławomir Biały (talk) 11:45, 10 June 2011 (UTC)
OK, I understand. Thanks for explaining. Gabriel Rosser (talk) 13:00, 10 June 2011 (UTC)

## Layman

It's not difficult to give 'the gist' of what the dirac delta function is to the layman, but the first sentence introduces difficulties by using the term 'generalised function'. So no way can the layman get the gist. I will attempt to give the gist in a new first paragraph, which will be there by the time you read this. — Preceding unsigned comment added by Mal (talkcontribs) 09:17, 16 January 2012 (UTC)

What you added was incorrect. I have removed it. We should, of course, try to simplify the treatment as much as possible, but we shouldn't tell lies in the process. The term "generalized function" is technically correct, and yet it also conveys the right sense to a layman. Sławomir Biały (talk) 10:00, 16 January 2012 (UTC)
Most laymen would have no idea what a generalised function is. If you doubt this, try asking your nearest and dearest (if he/she doesn't have a science degree!) S/he might also have trouble with "function" - but s/he might understand the first paragraph of the "function" article.Mal (talk) 16:03, 17 January 2012 (UTC)
Well that's most unfortunate. The concept of a function is included in elementary school education throughout much of the world. But this is irrelevant. Obviously one can't hope to get anything out of the article without knowing what a function is. The Dirac delta function is a "generalized function" (forget that this has a technical definition for a moment). To someone who understands the concept of "function", we should expect that such a person also understands the implications of the English word "generalized". Sławomir Biały (talk) 16:41, 17 January 2012 (UTC)
The first sentence contains "The Dirac delta function, or δ function, is (informally) a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero". This is not true, the sinc(x/a)/a function being a counter example. I agree, we should not tell lies in the process, and I wonder if this is too big a lie. PAR (talk) 01:12, 17 January 2012 (UTC)
It's not a lie. The delta function is zero at every nonzero point. (More precisely, its support is the single point at the origin.) The counterexample of sinc(x/a)/a doesn't tend pointwise to zero, but it does tend weakly to zero away from the origin. The only "lie" here is one that hasn't been committed, namely the naive belief that the pointwise limit is relevant. Sławomir Biały (talk) 01:23, 17 January 2012 (UTC)
One way of conveying what the delta function is intuitively would be to refer to the way Dirac described it. This way we don't commit ourselves to mathematical accuracy, and also give an idea of the intuition behind it. I don't have Dirac's book in front of me but I assume he spoke of a function that's zero everywhere except the origin, is infinite at the origin, and integrates to 1. If we say anything of this sort we would have to attribute it explicitly to Dirac. Tkuvho (talk) 13:02, 17 January 2012 (UTC)
This description appears in numerous sources, including Dirac, Gelfand and Shilov, Vladimirov, etc. It doesn't need to be specifically attributed to anyone. It is commonly accepted that the delta function is a measure supported at the origin with total mass one. Sławomir Biały (talk) 13:07, 17 January 2012 (UTC)
What I was pointing out is that we could distinguish between initial pre-mathematical intuitions, on the one hand, and mathematical implementations (of which there are several, including "true" functions), on the other. If we choose to present the former in the lede, then there is no need to mention "generalized" since Dirac did not do it. Also, there is no need to suppress "infinite at the origin", since Dirac presumably mentioned it, and also because it helps to give an intuitive idea of the delta function. Tkuvho (talk) 13:12, 17 January 2012 (UTC)
The definition is on p.58 of TPQM, and Tkuvho is correct. Mal (talk) 16:03, 17 January 2012 (UTC)
I just checked this reference. It agrees exactly with how we describe it in the article. Nowhere does Dirac appear to say "equal to zero everywhere except the origin, where it is infinite". Moreover, Dirac is careful to avoid calling it a function. He calls it an "improper function". The standard term nowadays is "generalized function." Sławomir Biały (talk) 16:51, 17 January 2012 (UTC)

Is the argument over whether to use the term "generalized function" in the lede? I think it should stay. Even if you don't know the technical definition of a generalized function the term still gets across the idea that the delta is like a function in some ways, but not really a function. The business about the delta as a function on the hyperreals is obscure and should not be a consideration for the first few sentences of the intro IMHO.Holmansf (talk) 16:49, 17 January 2012 (UTC)

The title of this talk section is "Layman". To a layman, the idea that
${\displaystyle \lim _{a\rightarrow 0}{\frac {\sin(x/a)}{xa}}=\delta (x)}$
and
${\displaystyle \delta (1)=0}$
means that
${\displaystyle \lim _{a\rightarrow 0}{\frac {\sin(1/a)}{a}}=0}$
which is false, the limit does not exist. The error is in the statement ${\displaystyle \delta (1)=0}$. The delta function should NOT be thought of as necessarily zero at every nonzero point. To simply blow off the problem by saying "pointwise convergence is irrelevant" is sloppy. To say that its support is the single point at the origin is NOT the same as saying it is equal to zero everywhere else. If f(x)=0 except f(0)=1, then it has no support on the real line. This does not mean that f(x) is zero everywhere. The delta function is "code" for a more complicated concept. The delta function has no real meaning except when used inside an integral. To say it is supported nowhere except at zero immediately invokes the concept of integration. To say it is zero except at the origin does not. Its like talking about ${\displaystyle \lim _{x\rightarrow \infty }1/x=0}$. There is no such real number as infinity, the whole statement is "code" for a more complicated limiting process which involves only real numbers. PAR (talk) 16:53, 19 January 2012 (UTC)
This post really veers off from anything that might possibly concern a layman. Rigorous formal definitions are given in the appropriate section, for someone to whom such issues might be a worry. I agree that talking about pointwise properties of the delta function is simply "code" to mean where it is supported, but I think you'll find that (outside of very formalistic writings) it is a code that is generally used to talk about the support of a distribution, if not to rigorously define it. Sławomir Biały (talk) 17:05, 19 January 2012 (UTC)
Slawomir Bialy is correct - the issues you're discussing would only concern someone who has spent some considerable time thinking about the delta function, not a general layman. But since you brought it up, I believe the mistake in the above "layman's" argument is actually a misunderstanding of the limit. It is not a pointwise limit, but rather in the sense of distributions. One can also say for example ${\displaystyle \lim _{a\rightarrow \infty }e^{iax}=0}$, but this doesn't mean the distribution zero cannot be said to be pointwise equal to zero. Certainly any distribution can be evaluated pointwise in the complement of its Singular support, and for the delta function that is everywhere except the origin. Holmansf (talk) 18:42, 19 January 2012 (UTC)

I read through this entire section, and although I'm not quite sure what the disagreement is, I think part of it is due to confusing several distinct objects as if they were the same:

1. The δ function, a non-existent function on the real line which is equal to 0 everywhere except possibly x=0, and which integrates to 1 on the entire real line.
2. The δ functional (distribution), which as such is not a function on the reals, so we cannot even ask whether it is zero at, e.g., x=2.
3. The point mass probability measure on the reals located at the origin. Again, this is not a function, so we cannot ask whether it is zero at x=2.

The latter two bullets do not give a definition of the δ function; they give examples of things that do exists although the δ function does not.

By analogy, if I accept that dragons do not exist, but are simply fictionalized crocodiles, and I accept that crocodiles cannot breathe fire, I can still assert that dragons can breathe fire. Similarly, even though the δ function does not exist, it is still 0 everywhere except possibly the origin. — Carl (CBM · talk) 22:53, 19 January 2012 (UTC)

Yes, I think it would be useful to clarify what precisely is being debated. My understanding is that the original proposal was to rewrite the lede in some way that does not include the term generalized function. Personally, I think generalized function should be used there, but I don't like the use of the word "parameter" and would change the first sentence to:
"The Dirac delta function, or δ function, is a generalized function on the real number line that is zero everywhere except at zero, and when integrated over the whole line is equal to one."
Holmansf (talk) 00:45, 20 January 2012 (UTC)
I also think that is better than the "parameter" sentence. — Carl (CBM · talk) 03:29, 20 January 2012 (UTC)
Looks good to me. Sławomir Biały (talk) 03:31, 20 January 2012 (UTC)

I'm not forgetting about the layman - I'm just saying that to the layman, the three equations I wrote will seem inconsistent. Yes, the first limit equation is "code" for something more complicated, not to be interpreted as if δ was a function. The second equation is not code for anything, it is simply wrong, it assigns a specific value to the delta function. The third equation therefore does not follow. So how do we present this to the layman? I think we do that by saying that thinking of the delta function as zero everywhere, infinity at the origin, and the integral of the delta function is unity is a good way to think of the delta function to begin with, but that further study will show that this is only a first step. I think it is wrong to tell or imply to a layman that this is the final definition of a delta function. Then present the sinc function as a counterexample to illustrate that fact. Adding the term "generalized", which changes the meaning of "equal" is not something that we should expect a layman will immediately get. I'm in favor of saying something like this in the introduction:

The Dirac delta function, or δ function, is (informally) a generalized function depending on a real parameter. It may be thought of as being zero for all values of the parameter except at zero, at which point it is infinite, and its integral over the parameter from −∞ to ∞ is equal to one. This interpretation is useful for many situations, but not appropriate for all situations. The full definition of the delta function rests upon its behavior upon integration, and thinking of it as having a specific value as a function of its parameter is sometimes counterproductive.

This will not call upon a layman to understand and probably misinterpret the unmentioned subtleties of "equal" with regard to a generalized functions. PAR (talk) 06:13, 20 January 2012 (UTC)

Your proposal is likely to be more confusing to readers rather than less so, for the common objection that it talks around the subject without defining it (even informally). Moreover, I know of no reliable sources that are concerned with this issue. Indeed, Dirac includes no such equivocal language. Laurent Schwartz (commonly credited as the founder of modern distribution theory) introduces it in the same way as Dirac, as do Gelfand and Shilov (two more founders of the modern theory of distributions). Do you have sources of similar pedigree that justify rewriting the lead to avoid the particular bogeyman that you are worried about?
Moreover, it has already been argued by multiple editors that the lead sentence is not misleading. The simplest solution to the issue you pose is not to write ${\displaystyle \delta (x)=\lim _{a\to 0}{\frac {\operatorname {sinc} (x/a)}{\pi x}}}$. It is this expression that is misleading to a layman, since the limit involved is a weak limit rather than a pointwise limit. I hope it's clear to you that the delta function is not a pointwise limit of any sequence of functions. So the misleading thing here is to believe that it is. Furthermore, as a weak limit, it is zero away from the origin. So it's also misleading to suggest that it might not be. Sławomir Biały (talk) 11:50, 20 January 2012 (UTC)
This language has exactly the problem I was talking about. The full definition of the δ function is that it is zero everywhere except the origin, and integrates to 1. (Of course no such function exists.) The full definition of the δ distribution is based on how it integrates with test functions, but the δ distribution is not the same thing as the δ function. — Carl (CBM · talk) 12:23, 20 January 2012 (UTC)
My engineering texts agree with Carl, it's very simple: "an ideal impulse has finite area but "zero" duration (it occurs entirely within the interval 0 < t < 0+). It must therefore have "infinite" height. . . on one ground or another, impulses are specifically exempted from the category of "finite disturbances" in the spirit of Table VI, p. 202. So, for exactly the same reasons, are the derivatives of steps." Cannon 1967:217. I.e. if integrated from -infinity to +infinity the integral produces an ideal step-function at time t=0. And vice versa, the derivative of an ideal step function is the delta function. The one sided Laplace transform of an impulse function is 1. Bill Wvbailey (talk) 14:40, 20 January 2012 (UTC)
Ok, I am going to go out on a limb here, tell me where I go wrong. There is no such thing as a "delta function", the notation ${\displaystyle \delta (x)}$ has no meaning except inside of an integral, and whenever you see it alone in an equation, it is "code" for describing the behavior of that equation upon integration. The behavior of the delta function upon integration is ${\displaystyle \int _{-\infty }^{\infty }f(x)\delta (x)dx=f(0)}$ and thats the entire definition. Ok, thats a bit informal, better is the exact definition of the Dirac measure. Furthermore, the idea that the "delta function" has a particular value at a particular point makes no sense given this definition. When you say ${\displaystyle \delta (1)=0}$ you are misleading the layman into thinking that it is some kind of point-wise statement applying to some function, when in fact it is a limiting statement about the integral of ${\displaystyle \int f(x)\delta (x)dx}$ over an interval about 1.
When you say
${\displaystyle \lim _{a\rightarrow 0}{\frac {{\textrm {sinc}}(x/a)}{a}}=\delta (x)}$
you are really saying
${\displaystyle \int _{-\infty }^{\infty }\lim _{a\rightarrow 0}{\frac {{\textrm {sinc}}(x/a)}{a}}\varphi (x)\,dx=\varphi (0)}$
Now, I admit, there is trouble. This is certainly true for certain restrictions on ${\displaystyle \varphi (x)}$, like it has to be continuous almost everywhere, I think. So in that sense, the sinc definition of the delta function MAY be misleading if you are thinking of L^2 integrable functions. I say "may" because I don't know what happens if you admit L^2 integrable ${\displaystyle \varphi }$. Is it misleading for Reimann integrable functions? I don't know.
When you say that the sinc definition of the delta function is a "weak limit", I interpret that to mean that the above integral equation is true almost anywhere for any ${\displaystyle \varphi }$ in L^2. Is this what you mean? I also admit that I do not know whether that is true or not.
The bottom line is that the sinc definition may very well be misleading, but the statement that ${\displaystyle \delta (1)=0}$ is also misleading, to the layman. Its a statement about the support of the Dirac measure, which, as Carl says, is not the same as the delta function, which, as Carl correctly says, does not exist. PAR (talk) 06:28, 27 January 2012 (UTC)
I'm just not seeing it. The article says that the delta function "is zero everywhere except at zero", which is a true statement about the delta function. If you're worried that it isn't a function, it agrees with the zero function away from the origin in a standard way: see Distribution (mathematics)#Functions as distributions. As far as I can tell, the the equation you seem to have a problem with, ${\displaystyle \delta (1)=0}$, doesn't appear anywhere in the article. The article is very careful to not refer to the pointwise value of the delta function at all. In the second paragraph of the lead, it explains that we need measures to make the (admittedly informal) statement of the first paragraph rigorous. Sławomir Biały (talk) 11:35, 27 January 2012 (UTC)
I meant that to a layman, ${\displaystyle \delta (1)=0}$ is a special case of "is zero everywhere except at zero". That statement "is zero everywhere except at zero" is misleading to a layman. Rather than say it "is equal" to zero, say it may be thought of as being (pointwise) equal to zero, but that ultimately this idea is not productive. Then we can go into the more complicated measure theory meaning of "is equal". PAR (talk) 05:03, 28 January 2012 (UTC)
Surely this is no more misleading than referring to elements of L2 as "functions". Sławomir Biały (talk) 11:15, 28 January 2012 (UTC)
Certainly true. So let's not mislead the layman by making either statement.
I think of a function as being defined point by point, and I think of elements of L2 as functions with an equivalence class imposed on them. This equivalence is not the same as pointwise equivalence, and that is the whole problem here. A layman sees an equal sign and assumes pointwise equivalence. In the very first sentence of the article, the article uses "equals" in the sense of an L2 equivalence class, without ever explaining the difference to the layman, and that is what I see as misleading. PAR (talk) 10:06, 29 January 2012 (UTC)
The δ function is pointwise equal to 0 everywhere except at the origin, though. That is one of its two defining properties, after all. The first sentence of the article isn't about L2 equivalence, nor is it about the δ measure or δ distribution. The point of the first sentence is to give the definition of the δ function.
By comparison, imagine if we had an article on dragons that first said that a dragon is a flying lizard the breathes fire, and then went on to give examples of actual animals that might have inspired the myth of dragons. Of course there is not really a fire-breathing flying lizard, but we would still need to say that is what a dragon is. The δ function is a similar mythological beast. — Carl (CBM · talk) 12:37, 29 January 2012 (UTC)
Note that the fire-breathing variety of the delta function is definitely infinite at the origin. This should be mentioned in the lede. Tkuvho (talk) 12:57, 29 January 2012 (UTC)
Since that first sentence has three sources, I'd be interested to know what they say about that before changing the sentence that is cited to them. The first sentence of the overview gives a different idea of what the graph would look like. I don't think that the value at 0 is worth adding a second sentence to the lede, because it seems to be a somewhat delicate issue. But if the sources already in use say that the function is infinite then I would add it to the first sentence. — Carl (CBM · talk) 13:12, 29 January 2012 (UTC)
They don't. Sławomir Biały (talk) 13:22, 29 January 2012 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I don't really accept the whole "fire breathing" argument. Since no one seems to agree about anything here, we should at least be able to agree with what reliable sources have to say about the delta function. I've already pointed out, all of the sources I checked initially gave an informal definition that was some variant on "zero everywhere with integral one". Not one of them said "infinite at the origin" (as Tkuvho would like the article to say). Not one of them seemed to feel that this heuristic was misleading in the way that PAR thinks it is. I checked the following very high-quality sources: Dirac, Laurent Schwartz (discoverer of distributions), Gelfand-Shilov (discoverers of distributions), Bracewell (standard signal processing text), Arfken and Weber (standard text for engineers and physicists). Now, I think I have given very good reasons (to PAR) that the informal definition is not misleading. This is supported by the presence of the same informal definition in many many high quality sources. But "zero everywhere except at the origin, where it is infinite" is misleading and misses the key point altogether. Some sources, like Vladimirov (standard text for mathematical physics) emphasize this very issue. Sławomir Biały (talk) 13:22, 29 January 2012 (UTC)

Thanks for looking up the sources, I really appreciate it. I think the source of "infinite at the origin" may be that, as everyone here knows, if the function was finite at the origin, it would (more) clearly not have a positive integral, and because if we think of the function as representing an impulse it would have to have infinite magnitude and zero duration. Of course, as everyone here also knows, the definition for nonzero points already implies having a zero integral, which is why the naive δ function doesn't exist, regardless what the value at the origin is supposed to be. — Carl (CBM · talk) 13:36, 29 January 2012 (UTC)
ok, then go back to the sinc function problem. Its not pointwise equal to zero everywhere except at the origin, yet upon integration it behaves as a delta function for a certain set of pointwise test functions. As I understand the delta function, ${\displaystyle \int _{a}^{b}f(x)\delta (x)\,dx}$ equals f(0) if [a,b] contains zero, zero otherwise. That is the entire definition. Forget being zero everywhere and infinite at the origin. It has no real meaning as a pointwise function. You can profitably visualize it as that pointwise function sometimes, but that visualization will lead you astray in certain cases. Thats what we should tell the layman in the lead. The integral definition (more formally stated than above) will not, but thats something that needs to be developed. PAR (talk) 17:26, 29 January 2012 (UTC)
Slawik wrote above "But "zero everywhere except at the origin, where it is infinite" is misleading and misses the key point altogether. Some sources, like Vladimirov (standard text for mathematical physics) emphasize this very issue." What does Vladimirov say exactly about this issue? Tkuvho (talk) 13:22, 30 January 2012 (UTC)
Vladimirov first defines the delta function to be density corresponding to a point mass of mass one situated at the origin. He approximates by a sequence of indicator functions, and then computes the pointwise limit, which has total integral zero. That is, the (Lebesgue or improper) integral of the function
${\displaystyle f(x)={\begin{cases}0&x\not =0\\\infty &x=0\end{cases}}}$
is zero. So it is misleading in the extreme to define the delta function this way, since it is inconsistent with the requirement that the delta function have total mass one. For this reason, he emphasizes that the appropriate limit is not the pointwise limit, but the weak limit instead. Sławomir Biały (talk) 13:39, 30 January 2012 (UTC)
I think we all agree about the mathematical point. I was wondering about precisely what Vladimirov described as being "misleading in the extreme" if he indeed used such terminology. Tkuvho (talk) 13:47, 30 January 2012 (UTC)
His precise words were "The contradiction here shows that the point limit of the sequence ${\displaystyle f_{\epsilon }}$ as ${\displaystyle \epsilon \to 0}$ cannot be taken as the density ${\displaystyle \delta (x)}$". He arrived at this conclusion because the function that is "zero everywhere but the origin where it is infinite" has total integral zero, which of course contradicts the key property of the delta function. Sławomir Biały (talk) 14:15, 30 January 2012 (UTC)
Thanks. I found an even more useful quote in an article by Aguirregabiria et al: "Notice also that, contrary to the intuitive idea that delta(x) is infinite at the origin, a sequence converging to delta(x), that is, corresponding to *R g(u)du51 with R5(2,), will behave at the origin as g g(0), which may converge to 1, but also to 0, or even to 2." This is a good candidate for the lede: we can mention that, contrary to the intuitive idea that delta(x) is infinite at the origin, the nascent functions can be assigned arbitrary values there, and give a reference to Aguirregabiria et al. The 2001 article is entitled "delta-function converging sequences". Tkuvho (talk) 14:38, 30 January 2012 (UTC)
There may be a place lower in the article for that, but I think it goes into too much detail for the lede. The point of the lede is just to give a general summary, not to get into fine details. At the same time, we need to be careful to distinguish the δ function from the δ distribution; Vladimirov seems to conflate them, or to only talk about the δ distribution. — Carl (CBM · talk) 14:52, 30 January 2012 (UTC)
Carl, why are you being difficult? Being infinite at the origin is a common "fire-spitting" intuition about the delta function, and should be addressed in the lede. If it were not a common intuition, Vladimirov and Aguirregabiria would not have addressed the issue. Besides, some of the most obvious nascent functions will indeed display such behavior at the origin. Moreover, for every fixed "small" interval [-e,e], the supremum of the nascent function will necessarily tend to infinity. In this sense, the intuition is not so far off the mark. Tkuvho (talk) 16:53, 30 January 2012 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I don't think we should emphasize the "value" at the origin at all. The intuitive description should stand as it is, since this is how the vast majority of reliable sources describe the delta function. The fact that there are sources saying that the delta function is not infinite at the origin seems to further bolster this point that it should not be described this way. Sławomir Biały (talk) 19:27, 30 January 2012 (UTC)

In exactly the same spirit, we should not emphasize the "value" anywhere else either. Then we are left with the fact that there is no "delta function" and that every reference to a "delta function" is a veiled reference to the delta distribution. Then we are left with the problem of how to introduce the "delta function" to the layman. Again, I say we should describe it as zero everywhere, infinite at the origin, such that the integral of f(x) over [a,b] is zero if [a,b] does not contain zero, f(0) if it does. Then we can say this is an oversimplification that is a good way to start but can lead to trouble, and then go on to explain from there. PAR (talk) 16:19, 1 February 2012 (UTC)
This is not a bad summary, but it may be better to refer to the bit about "infinite at the origin" as "an intuition that is not entirely correct", which is sourceable. On the other hand, this isn't going anywhere as we have two editors supporting the proposal and two opposing. Perhaps someone else can comment. Given the difficulty of the material, the more we say about motivating intuitions, the better, even if ultimately some of them have to be given up. Remember, a concrete mathematical implementation is always an impoverished version of the original pre-mathematical insight, as Weyl put it. Tkuvho (talk) 16:27, 1 February 2012 (UTC)
Sorry to keep pointing this out, but this is not how reliable sources introduce the delta function. The "infinite at the origin" bit might be sourceable to poor quality sources, but the majority of high quality sources do not define it in this way at all. I think we should stick to what the best sources say, which is what Dirac says, which is what the article says. Sławomir Biały (talk) 11:43, 2 February 2012 (UTC)
I have added the following sentence, that should hopefully be a suitable compromise: "The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized point mass or point charge." Sławomir Biały (talk) 11:55, 2 February 2012 (UTC)
Thanks, that sounds fine. As far as reliability of sources is concerned, there is no reason to believe Aguirregabiria is a poor source and not a solid scientist. A brief check at mathscinet reveals a number of texts of his. Tkuvho (talk) 13:00, 2 February 2012 (UTC)
Does he have a book or something? I can't find a citation. The sources I have mentioned (Dirac, Schwartz, Gelfand-Shilov, Bracewell, Arfken-Weber) all have thousands of citations, and are considered to be canonical texts. From the Aguirregabiria source that I see in AJP, he seems to argue against the point that the delta function should be thought of as "infinite at the origin". Have I missed something? Sławomir Biały (talk) 13:10, 2 February 2012 (UTC)
No, you haven't missed anything. I have been precisely suggesting that we refer to an intuition that is not entirely correct according to this source (and I imagine others). I also mentioned that fact that if authors find it necessary to mention this point, it must be indeed a common intuition. Also, hyperreals delta functions do take infinite values at suitable infinitesimal points. Tkuvho (talk) 13:53, 2 February 2012 (UTC)

--- re the proposed "intuitive" wording, above, for non-mathematicians: Good point about Bracewell as a source. If you want a firm source, with an elegant, "intuitive" interpretation, here's what Bracewell 1965:69ff has to say. I haven't looked at this in 40 years, and yet it comes back as clear and in focus as the day I first read it as an engineering student; it will be highly intuitive to an engineer or physicist or a calculus student. Notice the notion of a "measuring equipment", and his definition dodges the problems of the simplistic infinite integral of a point at zero. The treatment is quite nuanced:

(He references, via a book by van der Pol and Bremmer, Hermite, Caucy, Poisson, Kirchoff, Helmhotz, Kelvin and Heaviside as "historical sources"):
"It is convenient to have notation for intense unit-area pulses so brief that measuring equipment of a given resolving power is unable to distinguish between them and even briefer pulses. This concept is covered in mechanics by the term "impulse". . . The notation δ(x), which was subsequently [subsequent to Heaviside for the derivative of the unit step function cf footnote 2 p. 69] introduced into quantum mechanics by Dirac3, is now in general use [3 P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed., Oxford University Press, Oxford, 1947]. The underlying concept permeats physics. Point masses, point charges, point sources, concentrated forces, line sources, surface charges, and the like are familiar and accepted entities in physics. Of course, these things do not exist. . . .However, the impulse symbol δ(x) [footnote refers to it as a "generalized function" or "symbol"] does not represent a function in the sense in which the word is used in analysis (to stress this fact Dirac coined the term "imporper function"), and the above integeral [the classical one in the article that everyone's been fussing about] is not a meaningufl quantity until some convention for interpreting it is declared. . . ." (page 69-70)

He then defines δ(x) as the limit, as τ-->0, of the integral from -infinity to +infinity with respect to x of: 1/τ times the rectangle function Π(x/τ). He draws this notion on page 72 with rectangle functions of decreasing width as 1/τ gets taller, shows how the step function's slope increases as τ --> 0. And then goes on to extend exactly the same notion to triangle functions, sinc functions, Gaussian function, plus some other more fussy oscillatory functions, each with slightly different properties (cf pages 69-74). Bill Wvbailey (talk) 18:15, 2 February 2012 (UTC)

Thanks for this. Elaborating on physical motivation is a very good idea, I think. I am a little less enthusiastic than you are about how effective the passage above is. He mentions some important physical applications, but does not really explain how they tie in with the Dirac delta. His explanation in terms of the integral is the usual one, with a partilar choice of (discontinuous) nascent function, which is already mentioned in the lede. Tkuvho (talk) 09:24, 3 February 2012 (UTC)

I hunted in the article for Bracewell's definition but not knowing what a "nascent function" means I guess I missed it. (The article is a bit overwhelming for someone not a mathematician). I realized after posting the above that the animation of the spike is the same as Bracewell's drawings. RE physical motivation for a "delta" function (a pulse so brief it cannot be measured) Bracewell gives it in the sentences that follow what I quoted above, and in a subsequent example of a low-pass filter; I add an example of plucking a wire:

" . . . Of course, these things do not exist. Their conceptual value stems from the fact that the impulse response -- the effect associated with the impulse (point mass, point charge, and the like) -- may be indistinguishable, given measuring equipment of specified resolving power, from the response due to a physically realizable pulse. It is then a conveinece to have a name for pulses which are so brief and intense that making them any briefer and more intense does not matter." (p. 69)

He then continues with an example of a lowpass filter whacked with a voltage impulse (standard operating procedure for engineers curious about the natural response of a system -- hit it with an impulse):

" . . .it is readily observable, as the applied pulses are made briefer and brier, that the response setles down to a definite form. It is also observable that the form of the response is then independent of the input pulse shape, be it rectangular, triangular, or eve a pair of pulses. This happens because the high-frequency components, which distinguish the different applied pulses, produce negligible response. . . . the details [of the pulse] are irrelevant; it is necessary only that they be brief enough. Since the response may be scrutinized with an oscilloscope of the highest precision and time resolution, we must, of course, be prepared to keep the applied pulse duration shorter than the minimum set by the quality of the measuring instrument. The impulse symbol enables us to make abbreviated statements about arbitrarily shaped indefinitely brief pulses." (p. 71)

Finally (tho this is probably in the article, somewhere), he presents the Maclaurin series of the function e^(-1/x)H(x)(where H(x) is the "nascent function" above), and he claims (without a demonstration, actually) that all the terms vanish except "the remainder" that does not go away -- it contains all the "information", as it were.

This was just an example of impulse "functions" in engineering, what's important is the above: Finally, RE convolution integrals and natural responses of systems: another physical explanation of the lowpass filter hit by an impulse is that the impulse contains all frequency components with amplitudes of 1 (like white noise, only not persistent) or in real life, usually a gaussian distribution rolling off slowly to 0 at infinity (pink noise). Thus an impulse sounds like a click. Whatever the system's natural response is, the convolution of the impulse with the (unknown) response will reveal it. This is especially useful for resonant circuits: Example: Guitarists and pianists rely on this fact -- that you hit or pluck a string (in its center) and it vibrates afterwards with its natural (fundamental) frequency. The first major engineering project I worked on was to "pluck" (impact) a wire using an electromagnet and a capacitive discharge (i.e. an impulse). It didn't work for short wires without sending a pulse so intense it ran the danger of exploding the coal mines where this equipment would be used -- solution: lots of little pulses spaced out at different intervals to create a spread-spectrum. (I can source this, if necessary). Anyway, those are the sorts of physical motivations behind "impulse functions". Bill Wvbailey (talk) 15:25, 3 February 2012 (UTC)

I'm not sure how much of this belongs in this article, since there is already an article impulse response (that would actually benefit from some attention in my opinion). Whether this article should mention the impulse response more prominently than it does already, I don't really have an opinion on. Sławomir Biały (talk) 15:47, 3 February 2012 (UTC)

I struck the above as a distraction. The question is whether further development along Bracewell's line of attack would help a layman: RE concerning the idealization of the nature of events below the "resolution of instrumentation", ie the discovery that the shape of an "event" (e.g. the exact mathematical equation) doesn't matter beyond a certain "resolution" because no instruments can detect exactly what's happening. Is this idealization just lucky heuristic, a discovery about the behavior of the world, not provable, or is it derivable/provable from deeper physics? The article does poke at the idea with the decent example of the baseball hitting the bat, but it's not very prominent or developed, and nothing in the pertainent paragraph is sourced (now we have a pinpoint source). Maybe all this paragraph needs is a tidbit re instrumentation resolution and the reference to Bracewell. The historical is not bad (the source mentioned there -- van der Pol -- is the same as noted in Bracewell). It just needs something more specific about why Dirac up with the notion. Bill Wvbailey (talk) 22:58, 3 February 2012 (UTC)

Indeed Bracewell's heuristic is quite elegant, but as far as the lead goes, his view doesn't really fit in with how the article is currently written. Maybe a paragraph in the Overview section would be a good place to start. In the long run, a well thought-out "Physical motivation" section is definitely lacking. Regarding Dirac, in my view he was a mathematician first and a physicist second, so his view as expounded in TPQM is more like that of the early parts of the article in its present form, although there have obviously been subsequent refinements to Dirac's ideas that deserve elaboration. Sławomir Biały (talk) 03:03, 4 February 2012 (UTC)

RE augmenting the "Overview" and then adding a "Physical motivation section": I agree.

RE slightly different motivation in free-body mechanics: My old physics text that introduced us to "particles in a box" i.e. quantum mechanics has nothing about Dirac. My classical physics text (Resnick and Halliday 1966:214) gives a brief discussion of impulses (using the baseball and bat analogy) that is somewhat similar to Bracewell; but with application to free-body mechanics their motivation for use of a δ-"function" is slightly different: "During the collision we can safely ignore [gravity] in determining the change in motion of the ball; the shorter the duration of the collision the more likely this is to be true".

RE generalized functions: this is discussed nicely in Bracewell p. 87ff.

RE references: In my communications-theory text (Carlson 1968:44ff) the definitions of the δ-"function" are similar to those of Bracewell, but written differently to include 4 possible functions in the limit as candidates: the gaussian, the rectangular function, the sinc function and the sinc2 function. Carlson references Bracewell 1965 and Lighthill 1958 [see below]. Toward the end of the chapter he again discusses the theoretical behind the heuristic (cf p. 88):

"A satisfactory mathematical formulation of theory of impulses has been evolved along these lines [i.e. limit as tau --> 0 of the integral that I wrote above] and is expounded in the books of Lighthill1 and Friedman2. Lighthill credits Temple with simplifying the mathematical presentation; Temple3 in turn credits the Polish mathematician Mikusinski4 with introducing the presentation in terms of sequences in 1948. Schwartz's two volumes5 on the theory of distributions unify "in one systematic theory a number of partial and special techniques proposed for the analytical inerpretation of 'imporper' or 'ideal' fucntions and symbolic methods6. // the idea of sequences was curernt in physical circles before 1948, however7.
• 1: M. J. Lighthill, "An introduction to Fourier Analsys and Generalised Functions", Cambridge University Press, Cambridge UK 1958.
• 2: B. Friedman, "Principles and Techniques of Applied Mathematics," John wiley & Sons, New York, 1956.
• 3: G. Temple, Theories and Appliations of Generalized Functions, J. Lond. Math. Soc., vol. 28, p. 181, 1953.
• 4 J. G.-Mikusinski, Sur la methode de generalisation de Laurent Schwartz et sur la convergence faible, Fundamenta Mathematicae, vol. 35, p. 235, 1948.
• 5 L. Schwartz, "Theorie des distributions," vols. 1 and 2, Herman & Cie, Paris, 1950 and 1951.
• 6 Temple, op. cit. p. 175
• 7 B. van der Pol, Discontinuous Pheonmena in Radio Communication, J. Inst. Elec. Engrs, vol. 81, p. 381, 1937.
• 8 Schwartz, op. cit. p. 22.

I am curious about Dirac's motivation. I'm thinking it has to do with sampling theory and the distortions caused by the "aperature effect", when applied to particle waves (cf Carlson 1968:285). I'll sign off for now re "Layman"; maybe someday I'll come back to it, or maybe somebody ambitious can use some of the above. Bill Wvbailey (talk) 16:52, 4 February 2012 (UTC)

Sorry to add to this lengthy and old section, but I think there is a mistake above: "Again, I say we should describe it as zero everywhere, infinite at the origin, such that the integral of f(x) over [a,b] is zero if [a,b] does not contain zero, f(0) if it does." I would replace "f(0)" with 1, since nowhere is the former claimed to be the definition of the Dirac delta, while the latter is claimed in multiple locations. And, concerning the original topic of this section, I am sympathetic to the non-mathematician, but starting with an integration-based definition is more confusing for such a person than starting with the more intuitive yet essentially erroneous point-wise "function" definition. David Spector (talk) 19:26, 26 July 2013 (UTC)

## real

The claim that "The Dirac delta is not a true function, as no function has these properties" is factually incorrect. It would be accurate to say that no real function has these properties. Tkuvho (talk) 14:18, 17 January 2012 (UTC)

Well, yes, we are talking about Lebesgue integrable functions from the reals to something. Making the codomain be the set of complex numbers would not make any difference. I don't see that the claim is incorrect, could you explain what you mean? — Carl (CBM · talk) 14:22, 17 January 2012 (UTC)
A Lebesgue integrable function f has a natural hyperreal extension *f by the transfer principle. There exist "true" hyperreal functions which, when integrated against *f, will have the effect of the delta function on f. Thus, the delta distribution can be represented by a genuine function, if one extends the domain. Slawek feels that elaborating on the hyperreal viewpoint in this early section is inappropriate, and he may be right. What I am discussing is a separate issue, namely, that the sentence as stated is incorrect: it is possible to represent the dirac delta by a function, though not over the real domain. Adding a one-word clarification would not be counter to wiki undue weight policies. Tkuvho (talk) 14:27, 17 January 2012 (UTC)
The construction claimed at PlanetMath using hyperreals [2] is also "factually incorrect", IMHO, because the Dirac delta must have the property that it is zero everywhere except at one point x=0, while the function constructed on PlanetMath is nonzero on some non-zero values. In particular, if the function constructed there is integrated on ${\displaystyle (-\infty ,0)\cup (0,\infty )}$ (the hyperreal line minus exactly one point) the answer must still be 1, but it should be 0 if we were integrating the Dirac delta function. So at best we can represent some of the properties of the Dirac delta using nonstandard methods, but we do not seem capture all of them. But this is a digression.
In any case the due weight for nonstandard analysis in this article is extremely low, and I agree with Slawek that we do not need to add "real" to the paragraph in question. Is there any example of a graduate-level analysis book that is not directly about nonstandard analysis that uses nonstandard analysis to treat the Dirac delta function? It might be worth mentioning the hyperreal analogy in a separate paragraph, though. — Carl (CBM · talk) 14:41, 17 January 2012 (UTC)
You are right, you can't have it zero everywhere, though you can arrange for it to be infinitesimal at every real point (except 0). What I was referring to is the defining property of the delta function, namely that it produces the value of f at 0 when integrated against f. Apart from the hyperreal issue, this is something that should be emphasized in the lede or at the very least early on in the article. I withdraw my claim that the current statement is factually incorrect: I was mentally substituting the latter property. Tkuvho (talk) 14:51, 17 January 2012 (UTC)
The next subsection, "as a measure", states: "no true function for which the property ${\displaystyle \int _{-\infty }^{\infty }f(x)\delta (x)\,dx=f(0)}$ holds." This claim does not appear to be correct, as discussed above. I would suggest adding "real" here. Tkuvho (talk) 14:56, 17 January 2012 (UTC)
We could make similar corrections to almost every article on real analysis (e.g. "there is no number that is greater than 0 and less than 1/n for every natural number n" and every other statement that relies on the Archimedean property). I don't think that is generally worthwhile; the context of these articles is real analysis, and in that context the statements are correct. Moving to the context of hyperreals changes the meanings of all the symbols, so that they no longer mean what they used to. Thus statements that start out correct can become incorrect.
By analogy, in an article on topology, we could say that the rational numbers have ${\displaystyle 2^{\aleph _{0}}}$ subspaces, even though our article on locale theory says there are ${\displaystyle 2^{2^{\aleph _{0}}}}$ sublocales of the rational numbers. The statement in topology is not trying to talk about locale theory, though; statements in topology that are correct might nevertheless become incorrect if they are re-interpreted to be statements about locale theory. — Carl (CBM · talk) 16:31, 17 January 2012 (UTC)
Unlike the theory of real numbers per se, there is nothing about the Dirac delta or any of its numerous applications in physics that stipulate that it must be handled in the context of the real numbers specifically. Statements about cardinality are certainly about the real numbers. Statements about the Dirac delta are not. Certainly there weren't to Cauchy. Tkuvho (talk) 17:58, 17 January 2012 (UTC)

I don't think it would be out of line to add a subsection to the definitions section about this hyper real construction. However, I don't think it should be overly stressed. What do you think of a wording like this: "The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties. Most sources rigorously define the Dirac delta function as either a distribution or a measure. It is also possible to construct the delta function as a function on the hyper reals" I would shy away from the wording "The Dirac delta function is not a real function ..." as I think most people would parse this to "The Dirac delta function is not a real valued function ... ." You could then add a section about the hyper real construction after the other definitions. Holmansf (talk) 17:13, 17 January 2012 (UTC)

I think Carl makes a convincing case that the hyperteal construction does not belong in the Definitions section, since it's not actually a definition of the same thing. Perhaps it could be mentioned at the end of the article as a related construction though. Sławomir Biały (talk) 17:25, 17 January 2012 (UTC)
I certainly think mentioning the hyperreal construction in a section at the end is worthwhile, as it is an interesting result. I also like the wording suggested by Holmansf for the lede. I would also try to avoid the wording "true function"; "function in the traditional sense" is much more clear to me. — Carl (CBM · talk) 18:12, 17 January 2012 (UTC)

### Break

The infinitesimal section currently refers only to a bibliography by Yamashita. Is anyone willing to look at the original papers by Todorov to see what he actually says? Sławomir Biały (talk) 17:10, 19 January 2012 (UTC)

Which Todorov paper are you referring about? There is a paper of his in AMM as I recall in the 1990s but there were already several articles about the hyperreal Dirac delta by that time in the literature. Tkuvho (talk) 08:07, 20 January 2012 (UTC)
I mean the ones referenced by Yamashita. He's a bit vague on the details, but my impression is that others had only "approximated" distributions by smooth hyperreal functions, while Todorov was able to eliminate the "infinitely near" relation. Anyway, some kind of proper clear reference is highly desirable, whether it be to Todorov or someone else. Sławomir Biały (talk) 11:16, 20 January 2012 (UTC)
Actually, I take it back. You reference the bibliography inline, but there is another paper by Yamashita appearing in the footnote that contains more details. Sławomir Biały (talk) 11:19, 20 January 2012 (UTC)
Doing it "up to an infinitesimal" is already in Cauchy! You can start with any bump function, and use a sequence of rescalings to obtain an internal hyperreal function which evaluates to F(0) up to an infinitesimal at every real continuous function F. I recall there was a paper by a japanese in the 1960s, Giorello in the 1970s, and others before Todorov. Actually, it is surely already in Abraham Robinson, since the implementation of the Dirac delta is one item that Heyting singled out as a particular achievement of non-standard analysis. Tkuvho (talk) 12:58, 22 January 2012 (UTC)
That may be so, but clear inline citations should be added to the relevant section then. The current reference to an incomplete bibliography that references none of what you just mentioned seems poorly selected. Sławomir Biały (talk) 01:56, 23 January 2012 (UTC)
I have to revert myself again. If one starts with a bump function f with compact support and rescales it to obtain a sequence of "nascent" delta functions <f_n>, then the associated internal function [f_n] will indeed vanish at every nonzero real point, as I originally claimed. Therefore we obtain a function defined on the hyperreals which vanishes at every real point such that, when integrated against, will have the effect of the Dirac delta function. This was my original remark: it is not entirely correct to assert that "there is no function with these properties". It is only correct to assert that there is no real function with these properties. Sorry about the confusion. Tkuvho (talk) 15:16, 1 February 2012 (UTC)
The hyperreal delta function thus constructed will take infinite values for suitable infinitesimal values of the input. Tkuvho (talk) 15:17, 1 February 2012 (UTC)
Note that Yamashita does cite an earlier source: G. Takeuti, Proc. Jpn. Acad. 38. 414 (1962). Also, non-standard delta functions are indeed mentioned in Robinson's 1966 book. Tkuvho (talk) 15:03, 15 March 2012 (UTC)

## Error under Representations of the delta function - Probabilistic Considerations

The formula under Representations of the delta function - Probabilistic Considerations is wrong. The formula given is

${\displaystyle \eta _{\varepsilon }(x)=\delta _{n}(x)=2n\ {\textrm {rect}}(nx/2)={\begin{cases}2n,&-{\frac {1}{n}}

Which is correctly copied from the reference (Aratyn & Rasinariu 2006), but the book has a mistake in, they say that 2n(1/n - (-1/n)) = 1 which is just wrong, it's 4. The correct formula should just be

${\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\varepsilon }}\ {\textrm {rect}}\left({\frac {x}{\varepsilon }}\right)={\begin{cases}{\frac {1}{\varepsilon }},&-{\frac {\varepsilon }{2}}

I don't know if this counts as "original research" though so I thought I should mention it here before adding it... — Preceding unsigned comment added by 129.11.69.169 (talk) 12:15, 6 March 2012 (UTC)

It's fine by me if you want to make the change. It also seems strange that we would change to a sequence involving n rather than ε. It's not a big issue, but if you can find another reference that would be ideal. Sławomir Biały (talk) 12:25, 6 March 2012 (UTC)
I just noticed Saichev and Woyczyński mention it - they're already listed in the references! Up to a factor of 2 anyway, but that's consistent with ηε(x) = η1(x/ε)/ε — Preceding unsigned comment added by 94.192.37.97 (talk) 10:08, 7 March 2012 (UTC)

8th of August 2012

Hello, in the Definition it is said, that the Heavyside step function is 1 for ${\displaystyle x\geq 0}$, otherwise 0. below that, the characteristic function is said to be in the range ${\displaystyle (-\infty ,0]}$.

Hasn't this to be ${\displaystyle [0,\infty )}$

I'm a bit confused, since it appears several times, so it might be on purpose.

Thank you for clearance.

Best

Phil. — Preceding unsigned comment added by UnameAlreadyTaken (talkcontribs) 22:23, 7 August 2012 (UTC)

Where do you see the characteristic function of (-∞,0]? Sławomir Biały (talk) 00:47, 8 August 2012 (UTC)

## The frequency domain

I would like to see some specifics on the Dirac Delta in the frequency domain. The mere substitution of "x" by "f" in the current definition formulae would be misleading. This clarification is important since some other articles depend on it. For instance, the statement The Fourier transform of a Dirac comb is also a Dirac comb here.MaskedAce (talk) 03:31, 30 September 2012 (UTC)

The Fourier transform is already covered in detail in the relevant section. It's not clear what you specifically believe is inadequate in the present treatment. Sławomir Biały (talk) 21:24, 30 September 2012 (UTC)

In the equation that follows the expression "The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed": When I naively try to do some calculus I obtain ${\displaystyle \int _{f=-\infty }^{\infty }e^{2\pi ift}df={\biggl [}{\frac {e^{2\pi ift}}{2\pi it}}{\biggr ]}_{f=-\infty }^{\infty }={\biggl [}{\frac {\cos {2\pi ft}+i\sin {2\pi ft}}{2\pi it}}{\biggr ]}_{f=-\infty }^{\infty }={\biggl [}{\frac {\sin {2\pi ft}}{2\pi t}}{\biggr ]}_{f=-\infty }^{\infty }=\lim _{f\to \infty }{\frac {2\sin {2\pi ft}}{2\pi t}}={\Bigg \{}{\begin{matrix}\lim _{f\to \infty }2f&{\textrm {as}}&t\to 0\\\in [-1,1]/(\pi t)&{\textrm {if}}&t\neq 0\end{matrix}}}$

Is the Dirac pulse amplitude defined to be ${\displaystyle 2\infty }$ so to speak? My question is if there is a reason that the definition doesn't contain a compensating factor of 1/2. I simply just don't understand and I believe that the article would benefit much if someone had any idea on how to explain the above in a pedagogical sense. Geo39geo (talk) 13:23, 12 February 2018 (UTC)

The value of the function ${\displaystyle \textstyle {\frac {2\sin {2\pi ft}}{2\pi t}}}$ at 0 is roughly the height of the hump; but did you think about the width of the hump? The integral of this function (over the whole real line) is equal to 1 (rather than 2), since ${\displaystyle \textstyle \int _{0}^{\infty }{\frac {\sin at}{t}}\,dt={\frac {\pi }{2}}}$ for all ${\displaystyle a>0.}$ Boris Tsirelson (talk) 18:53, 12 February 2018 (UTC)
Aha ok, I get it! This makes sense. Thank you very much for the succinct and fast reply. Geo39geo (talk) 22:10, 13 February 2018 (UTC)

## Elementary representation of the function

Is this function not easily representable as the elementary function ${\displaystyle {\frac {0}{x}}}$? Pokajanje|Talk 04:22, 18 May 2013 (UTC)

No, it's not. The delta function, as noted in the article, must be thought of as a distribution rather than a function. But see distribution (mathematics) for how to multiply a distribution by a scalar. In particular, zero times any distribution gives the zero distribution. So as a distribution ${\displaystyle 0/x=0}$, even though as a function the left-hand side is not defined at 0. Sławomir Biały (talk) 11:58, 18 May 2013 (UTC)
No, it's not. Your proposed function is everywhere zero except at x=0, where it is undefined. In most contexts, the expression ${\displaystyle {\frac {0}{0}}}$ can have any value at all, and so it is not a function. To clarify, the Dirac delta is properly defined only in the context of integration. Thus, the Dirac delta should be thought of as similar in usage to an infinitesimal like ${\displaystyle dx}$. It has no existence by itself in the conventional sense. The Dirac delta is a generalized function and not a function at all. David Spector (talk) 19:45, 26 July 2013 (UTC)

## not ${\displaystyle x\delta '(x)f(x)=-\delta (x)f(x)}$

this does not seem to be correct! Have a look at: http://www.wolframalpha.com/input/?i=integrate+from+-inf+to+inf+f%28x%29*x*delta%28x%29%27%3D-integrate+from+-inf+to+inf+f%28x%29*x*delta%28x%29 . A restriction has to be made!!!!!--92.202.98.107 (talk) 19:32, 28 November 2013 (UTC)

You made an input error in Wolfram Alpha. Please don't continue to insert your addition. Sławomir Biały (talk) 20:41, 28 November 2013 (UTC)
I do not see an input error. Show me what is wrong. In my opinion the "property" does not hold for an "additional test function!". You can also try to apply partitial integration... i dont see why this property should hold in general... So now show me the claimed input error! (In my opinion there is none) — Preceding unsigned comment added by 92.202.98.107 (talk) 20:50, 28 November 2013 (UTC)

Wolfram showed that f(0)=0 if this property holds for all test functions f. This is for sure not true. So this claimed "property" allows no further test function. Please provide a source for your claim.--92.202.98.107 (talk) 20:56, 28 November 2013 (UTC)

Ok, then why would you think that what is written in the article implies that
${\displaystyle \int x\delta '(x)f(x)=-\int x\delta (x)f(x)}$
?? I agree that this expression is incorrect, but it's certainly not what the article asserts. Sławomir Biały (talk) 21:03, 28 November 2013 (UTC)
Good that you agree. For sure the article would state ${\displaystyle \int x\delta '(x)f(x)dx=-\int x\delta (x)f(x)dx}$ without a further restriction because because the two *real* properties just above this one are general! The pseudo property for the test function x arises from partial integration. Therefore it holds for the test function x. In some sense it is interesting (for example I just searched for something like that), but it should be clear that this (pseudo) property of the delta function is not general (it is not a real property of the delta function since it only holds for one specific test function - or the test functions that fulfill f(0)=0).--92.202.98.107 (talk) 21:14, 28 November 2013 (UTC)

0).

What your wolfram alpha link shows is that ${\displaystyle x\delta '(x)\not =x\delta (x)}$. I agree with this, and it is not inconsistent with the article, which asserts that ${\displaystyle x\delta '(x)=\delta (x).}$ Do you not see the difference? Sławomir Biały (talk) 21:16, 28 November 2013 (UTC)
I am soo sorry for the inconvenience!!!!! http://www.wolframalpha.com/input/?i=integrate+from+-inf+to+inf+f%28x%29*x*delta%28x%29%27%3D-integrate+from+-inf+to+inf+f%28x%29*delta%28x%29 --92.202.98.107 (talk) 21:26, 28 November 2013 (UTC)

Here is the diff for all my changes: https://en.wikipedia.org/w/index.php?title=Dirac_delta_function&diff=583710621&oldid=577041997 --92.202.98.107 (talk) 21:40, 28 November 2013 (UTC)  Done

## Application:Solving differential equations

The Dirac delta function is used to find the impulse response when taking a Laplace transform https://en.wikipedia.org/wiki/Impulse_response. — Preceding unsigned comment added by 66.37.66.96 (talk) 01:19, 21 July 2015 (UTC)

## Can we rename it to 'Dirac delta distribution' ?

We say all the time to the students that ${\displaystyle \delta }$ isn't a function, so it would be nice if the wikipedia article wasn't named... Dirac delta function ! I hope you agree with that ? Can someone do the renaming properly ?

78.196.93.135 (talk) 21:13, 13 July 2016 (UTC)

The guideline for article naming is WP:COMMONNAME. The term "Delta function" is much more common than "Delta distribution", as an immediately recognizable name for the subject of the article. Sławomir Biały (talk) 23:49, 13 July 2016 (UTC)
You are proving you are wrong, the most common in mathematics is of course "Dirac delta distriАbution" while in physics it is probably "Dirac delta function" (anyway Dirac is always mentioned). This is clearly a mathematical article, and letting Dirac delta function in the introduction will let it referenced at 1st on google. — Preceding unsigned comment added by 78.196.93.135 (talk) 17:39, 15 July 2016 (UTC)
The most common in mathematics is certainly not "Dirac delta distribution". For example the Springer Encyclopedia of mathematics article is also entitled "delta function". Indeed, most sources in mathematics as well as physics and engineering use the term "delta function" quite happily. The multivolume authoritative treatise by Israel Gelfand and Georgiy Shilov, for example, uses "delta function". Sławomir Biały (talk) 16:09, 16 July 2016 (UTC)
It appears the article should be renamed. The first sentence of the article even clarifies that the Dirac delta distribution (or whatever we should call it) is not technically speaking a function. Somebody should find a good mathematical source that references it and rename the article as it is currently misleading. Jbeyerl (talk) 14:40, 16 July 2016 (UTC)
What is the basis for the proposed renaming? The subject of the article is almost universally known as the "delta function". Calling it "delta distribution" is very likely to be unfamiliar to most readets. Sławomir Biały (talk) 16:09, 16 July 2016 (UTC)
It seems to fail the precision guideline at Wikipedia:Article_titles because it purports that it is a function. Indeed as far as I know experts usually call it the "Dirac delta function" or "delta function", but their intended audience are typically other experts that know what they mean. Wikipedia should be more accessible to non-experts. The term is a colloquialism or perhaps even an idiom. What about this, leave the title the same but change the first sentence to "In mathematics, the Dirac delta function, or δ function, is a colloquialism used to describe the distribution on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line." Jbeyerl (talk) 21:57, 16 July 2016 (UTC)
Yes experts call it the "delta function", students call it the "delta function", engineers call it the "delta function", physicists call it the "delta function". That's precisely why the title of the article ought to be "delta function". It's the overwhelmingly widely accepted term for the subject of the article. WP:COMMONNAME. Sławomir Biały (talk) 22:45, 16 July 2016 (UTC)
The second paragraph of the article describes the sense in which the delta function is not a function. I don't think the reader is served by calling it a "colloquialism". Anyway, this lacks a reference. It is typically called a "generalized function" in the literature. This has the virtue of being true, well-sourced, and also clarifying the sense in which "function" is appropriate. As I said, the sense in which it is not a function are clearly explained in the second paragraph of the lead, with much more detail in the body of the article for those readers that wish to read further. Sławomir Biały (talk) 12:09, 17 July 2016 (UTC)
This article is titled function but is not about a function hence the very first sentence should make clear what it's actually about. I can't think of a better title, but as you even pointed me toward WP:COMMONNAME, it specifically says "Ambiguous or inaccurate names for the article subject, as determined in reliable sources, are often avoided even though they may be more frequently used by reliable sources." The best that seems doable in this case is clarifying what it means. Jbeyerl (talk) 13:41, 17 July 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Please read the second paragraph of the lead of the article. Please discuss and get consensus before reinstating these edits. You are "fixing" nonexistent problems with the article. It is made very clear that the delta function is a generalized function, and not a true function in the strict sense. We do this in a manner supported by very high-quality references, representing mainstream scholarship in the area. Sławomir Biały (talk) 14:27, 17 July 2016 (UTC)

The original point the anon editor pointed out brought out what does appear to be an issue. It's not clear to a novice that a generalized function is not a function. So the article reads as quite confusing to nonexperts. Jbeyerl (talk) 10:43, 18 July 2016 (UTC)
I think we should assume that a novice wishing to find out about the delta function will read more than the first sentence of the article. The entire second paragraph of the article concerns the delta function's lack of functionhood. Calling it an "idiom" is not helpful to a novice, and is borderline WP:OR because it is not supported by the cited sources, which include Gel'fand and Shilov, Dirac, and Schwartz. And I struggle to see how this edit, which is ungrammatical, clarifies anything. Finally, I have said to get consensus first, and you've not done that. Sławomir Biały (talk) 10:49, 18 July 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── The common name of the thing is the Dirac delta function, and we can't change that. Don't read it as two separate word but as one name. The Sierpinski sponge is not really a sponge. Do you mean we should we rename that article too? For the fine print, in another article it is referred to as a "formal device" with a well-defined distribution backing up its formal behavior rigorously. "Idiom" does not sound right. Could "formal device" be used in descriptive terms here? YohanN7 (talk) 11:56, 18 July 2016 (UTC)

I don't see anything wrong with the current formulation. "Generalized function" already signals that it is not quite a function. The second paragraph then elucidates in what sense this is not a function. Note that the vast majority of readers will not have a precise notion of what a function is (mathematically speaking) in the first place. Introducing woolly terminology like "idiom" or "formal device" will (at best) serve to add confusion, and are certainly not helpful.TR 12:29, 18 July 2016 (UTC)
For a precise description, "generalized function/distribution/continuous linear functional/whatever applies" is fine with me too of course. But the Dirac delta function is USED as a formal device. Few physicists and no engineers bother about what it really is, they just compute using the rules (which say it can formally be treated as a function) of the device. YohanN7 (talk) 12:43, 18 July 2016 (UTC)
I don't think "formal device" is appropriate for the first sentence. One thing that Schwartz notes is that the delta function was quickly adopted into the operational calculus (which we might call the "calculus of formal devices that engineers use"), and along with all of the methods of the operational calculus was viewed with suspicion by most of the mathematicians of the early 20th century. Sławomir Biały (talk) 12:51, 18 July 2016 (UTC)
I've added a little bit on this perspective to the second paragraph. Probably it could be improved. Sławomir Biały (talk) 13:03, 18 July 2016 (UTC)
I didn't mean to be used in the first sentence. It was just an idea, and now reading the lead (something I should perhaps have done in the first place), it is pretty much all there, especially after your edits. YohanN7 (talk) 13:05, 18 July 2016 (UTC)

My 2p as an uninvolved math editor. The term "delta function" is standard in the literature, so it's a good name for the article. "Delta distribution" is not as good because "delta function" can also refer, interchangeably, to the point-mass Dirac measure. I do think the lede should spend more time on the measure-theoretic representation, and not begin with a sentence about the distribution. It might be better to say that the delta function is a hypothetical function with certain properties; it turns out that such a function cannot exist, but similar objects do exist, are widely used, and are called the "delta function". Good luck, — Carl (CBM · talk) 14:57, 18 July 2016 (UTC)

The current second paragraph discusses the non-functionhood of the delta function. Is this what you mean? I don't see any reasonable way to pack the content of the second paragraph into the first paragraph. I think a basic rule of thumb is not to rush into things. It is not a problem if the first sentence misses some nuances. Sławomir Biały (talk) 15:32, 18 July 2016 (UTC)
I mean that, in my opinion, the first paragraph jumps the gun by mentioning the distribution interpetation but not the measure-theoretic one. The term "delta function" can be interpreted as a distribution or as a measure; the first paragraph instead says that it is a distribution. The section "Definitions" does a good job with both definitions, but then the "properties" section assumes we are back to the distribution interpretation, with no comment to that effect. — Carl (CBM · talk) 16:40, 18 July 2016 (UTC)
Well, strictly speaking the measure and the distribution are two different objects, and the term "delta function" really means the distribution rather than the measure. Indeed, there is a separate article Dirac measure. Really, the "distribution" is actually a Radon measure; but this isn't generally what people think of when they say "Dirac measure". They're thinking of the "geometrical" point mass at the origin, rather than its action on test functions obtained by integration. This should perhaps be clarified in the "Definition" section more than it already is. Sławomir Biały (talk) 16:59, 18 July 2016 (UTC)
This is the difference in our perspectives. In my mind, the term "delta function" refers ambiguously to both the distribution and the geometrical point-mass measure, by referring directly to neither, allowing the reader to interpret it either way in many situations, such as in the integral ${\displaystyle \int _{-1}^{1}\delta (x)\,dx=1}$. In some contexts "delta function" would almost always mean the distribution, but in others it would almost always mean the geometrical measure. If the goal of this article is to describe only the distribution, then I suppose it might make more sense to rename it to a different title and make the delta function article a disambiguation page. — Carl (CBM · talk) 20:27, 18 July 2016 (UTC)
I don't think this is a view that is widely held in the literature. I have not seen in the literature a definition resembling the following: "The delta function is the measure that assigns to a set A of real numbers the integer 1 if 0 is in A and 0 otherwise". One would always say "delta measure" (or "point measure", "unit mass", etc.) in that case. Typically, "Delta function" refers exclusively to the distribution, although I'm willing to be proven wrong by some references of a suitably high quality. Sławomir Biały (talk) 20:52, 18 July 2016 (UTC)
I guess, you could find something like this: "In particular, the delta function is evidently positive, and therefore, it is in fact a measure (in contrast to its derivative)". Boris Tsirelson (talk) 20:58, 18 July 2016 (UTC)
True enough, but this does not really support CMBS contention that distributions are over-emphasized in the article at the expense of measures. It is a theorem that the delta function is the distribution associated to a measure. But one does not usually refer to the measure itself (a function on a sigma algebra of a specific kind) as "delta function". The article discusses distributions and measures. I think giving measures more significance along the lines of what CBM is suggesting does not seem in line with NPOV. Sławomir Biały (talk) 21:21, 18 July 2016 (UTC)
Locally finite (signed) measures are (canonically) embedded into Schwartz distributions, as (for instance) integer numbers are (canonically) embedded into real numbers. Who bothers to specify, is 25 treated as integer or real (unless we do numerics)? Boris Tsirelson (talk) 20:47, 18 July 2016 (UTC)
A quote: “Like the alligator pear that is neither an alligator nor a pear and the biologist’s white ant that is neither white nor an ant, the probabilist’s random variable is neither random nor a variable.” S. Goldberg “Probability: an introduction”, Dower 1986, p. 160. (Alligator pear = avocado; white ant = termite.)
Such is the life. Indeed, also "delta function" is not a function. Boris Tsirelson (talk) 20:00, 18 July 2016 (UTC)

In addition: "finite measure" is a measure, but "signed measure", "vector measure" and "finitely additive measure" are (generally) not measures. Boris Tsirelson (talk) 20:41, 18 July 2016 (UTC)

This is a maths article, and I'm quite sure all the people who approve Dirac delta function are indeed not mathematicians, but engineers or physicists instead. In maths we say all the time that ${\displaystyle \delta }$ is a DISTRIBUTION, and its name is depending on the context : Dirac delta or Dirac delta distribution. In french, they renamed it to fr:Distribution_de_Dirac Please also understand it is different to the Kronecker delta, a function this time. This is addressed in particular to you said your opinion, let the other participate too, tks. 78.196.93.135 (talk) 03:52, 7 August 2016 (UTC)

You are mistaken. CBM and myself are both mathematicians. I wrote most of the article, that you think is a "maths article". But it's not really important what we think. It's important what reliable sources call it. The Springer Encyclopedia of Mathematics calls is the "delta function". Israel Gel'fand and Georgiy Shilov call it the "delta function" in their authoritative treatise "Generalized functions". Laurent Schwartz calls it "la function de Dirac". Nelson Dunford and Jacob Schwartz call it the δ-function in their "Linear operators" text. These are all mathematical works, written by the 20th century's most distinguished authorities on the theory of distributions. So, they cannot simply be dismissed as the works of physicists and engineers. Finally, the neutral point of view policy strongly urges against dismissing other points of view in this discussion. So, to get a crude estimate of how common "delta function" is versus "delta distribution" in the literature, we can compare Google scholar hits: "Dirac delta distribution" versus "Dirac delta function". Thus "Dirac delta function" is almost 20 times more common than "Dirac delta distribution" in the scholarly literature. The scholar search without the word "Dirac" shows a similar 20-fold preference of "function" over "distribution": "delta distribution" versus "delta function". While this certainly has engineering and physics mixed with mathematical sources, I've already demonstrated that even exceptionally high mathematical sources quite happily call it the "Dirac delta function". Sławomir Biały (talk) 11:30, 7 August 2016 (UTC)
you said your opinion : follow the wikipedia guideline, rely on old textbooks, some of them written before distributions had been even invented (~1950). Whereas our is : this is a maths article, and in mathematics the un-official guideline is to use the technical term whenever it is possible, since maths articles are mainly read by maths/science students. Here, the problem is that students discover ${\displaystyle \delta (x)}$ with the Fourier series, 2-3 years before they study the distributions, so there are 2-3 years while it is very possible they think that ${\displaystyle \delta (x)}$ is a function, and this is very error prone (see any maths forum, you'll see what I mean). The last but not the least argument, is that wikipedia is becoming the standard in science, i.e. the name you choose today for a wikipedia article will be the name tomorrow's teachers will use in class. 78.196.93.135 (talk) 23:26, 16 August 2016 (UTC)
I understand that your opinion is that the article should be renamed to "Dirac delta distribution". However, opinions of random people on the internet carry very little weight, and Wikipedia is not a democracy. We go by reliable sources, which I have given several, and Wikipedia guidelines, which clearly say we should use the more common term and I have demonstrated that sources overwhelmingly favor "function" in this context. Unless arguments are based on our policies and guidelines, they are likely to be disregarded. Sławomir Biały (talk) 23:42, 16 August 2016 (UTC)
Also, for some reason IP wants to disregard from consideration books written by the originators of the theory of distributions (Schwartz, Gel'fand). A much more recent work is Michael E. Taylor's three-volume work "Partial differential equations", wherein he refers to the delta function. Sławomir Biały (talk) 12:01, 19 August 2016 (UTC)
@ Ip: If you look at the top of this page you'll see that this article is within the scope of physics and engineering, hence not a pure math article. So even if your arguments had been correct (and they aren't) for a pure math article, this isn't such an article. YohanN7 (talk) 08:41, 17 August 2016 (UTC)

Obligatory google ngram test for "Dirac delta function" vs. "Dirac delta distribution" for a WP:COMMONNAME discussion. No contest in this case.TR 10:48, 17 August 2016 (UTC)

Ok then look also at the results for "dirac delta"+"is not a function" compared to "dirac delta" you'll find 70000 vs 500000 results. In other words, your choice is error prone, and a very bad idea from the scientific point of view. I'll conclude by saying wikipedia is the 1st result on google, so the name on wikipedia is the name that teachers and students will use for the 30 years to come...
And @YohanN7 : if you want to say that my arguments aren't correct, you have to explain why (and from what point of view). In particular you have to explain if you explain why the Dirac delta is a distribution and not a function (which is quite a good reason for not naming this article Dirac delta function). 78.196.93.135 (talk) 02:33, 14 November 2016 (UTC)
Riemannian metrics aren't metrics. They are Riemannian metrics. No problem there, right? There is no particular inherent problem with delta functions not being functions either. It is simply what the things are called. YohanN7 (talk) 09:58, 14 November 2016 (UTC)
The article very carefully explains why the delta function is not a function. It would need to do this whether the title of the article was "dirac delta function" or "dirac delta distribution", for two reasons. First, because it is overwhelmingly called "delta function" in the literature, it is necessary to convey this because of readers arriving here by typing "delta function" into the search bar. Secondly, this is a notable aspect of the distribution that a large number of high quality sources discuss. Furthermore, if we were to change title to "Dirac delta distribution", most readers would arrive here by a redirect, and we would need to spend more time, not less, explaining that the subject of the article is usually called the "Dirac delta function", in addition to discussion about why it is not a function. Sławomir Biały (talk) 11:14, 14 November 2016 (UTC)
Note that your comments make no sense what-so-ever with regard to ngram (For example, ngram does not support a "+" join, and returns only relative values, not absolute one). I think you are mistaking it with a standard google search. Ngram simply compares the occurrences of specific phrases in the printed literature over time. It is a great tool for determining which of alternative phrases (or spellings of the same word) is more common (and how this may have changed in recent history).TR 11:46, 14 November 2016 (UTC)
The "not a function" also refers to that it is not a real-valued function of a real variable. I think it is correct to say that it can be modeled as a perfectly acceptable (set-theoretic) function from the reals to the extended reals. The extended reals can certainly be modeled as a set. Begin with defining positive infinity to be any set except a real number (reals can be modeled as sets), e.g. the set of the real numbers itself, which certainly isn't a real number, thus put ∞ ≡ ℝ. (Define negative infinity if desired, define order, topology and algebraic operations in the nearly obvious ways if desired.) The result is that the delta function is a function. This is not very useful, but is i m o worth mentioning here on the talk page since so many get so involved in this issue. YohanN7 (talk) 12:36, 15 November 2016 (UTC)
That's actually not the case. The resulting extended real function would be measurable, and it would have Lebesgue integral equal to zero, not one. Sławomir Biały (talk) 13:05, 15 November 2016 (UTC)
You miss my point. It is a function, but it isn't very useful. Obviosuly, the Lebesgue integral with Lebesgue measure isn't what we have. Just a set-function. YohanN7 (talk) 13:10, 15 November 2016 (UTC)
But this set function would in no way model the delta function. That's a "function" whose integral is one that is equal to zero away from the origin. There is no function (in the set theoretical sense) with this property, extended real-valued or otherwise. Sławomir Biały (talk) 14:53, 15 November 2016 (UTC)
Point taken. But equally well, can't you put the quotation marks around "integral" (or call it a Dirac delta integral)?. YohanN7 (talk) 15:20, 15 November 2016 (UTC)

@YohanN7: Here in the comments, example of a mathematical student who thinks that the Dirac delta is a function because you on wikipedia don't understand what I'm saying. The introduction of the article is very unclear : "From a purely mathematical viewpoint, the Dirac delta is not strictly a function". Why not simply say that the Dirac delta is NOT a function ? 78.196.93.135 (talk) 21:26, 30 May 2017 (UTC)

(re to 78.196.93.135) Yes, why not? Done. YohanN7 (talk) 07:15, 31 May 2017 (UTC)
I'll take Gelfand, Shilov, and Schwartz over random anonymous people on the internet, thanks. The second paragraph already elaborates on the sense in which it is not a function, and how to make the idea of the Dirac delta function rigorous. Anyone who reads the paragraph and gets the idea that the Delta function is actually a proper mathematical function has issues with reading comprehension that are well beyond our scope to fix. Sławomir Biały (talk) 22:04, 30 May 2017 (UTC)

Obviously you didn't understand Gelfand, Shilov, and Schwartz. How do you explain everything I wrote ? How do you explain that many students think it is a function ? How do you explain I'm coming here to complain ? You have issues understanding something as simple as wikipedia is used as a main mathematical ressource by millions of students, and we have to make it clear that ${\displaystyle \delta }$ is not a function. Also I said it 5 times : we understood what you had to say, no need to repeat it. And when reading YohanN7's comment about ${\displaystyle \delta }$ being a function from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} \cup \{\infty \}}$ I'm really worried. No ${\displaystyle \delta }$ is not a function. It is a measure, a generalized-function or a distribution. The most useful definition in Fourier analysis is as the linear operator

${\displaystyle \int _{-\infty }^{\infty }\delta (t)\varphi (t)dt{\overset {def}{=}}\lim _{\epsilon \to 0^{+}}\int _{-\infty }^{\infty }{\frac {1_{|t|<\epsilon }}{2\epsilon }}\varphi (t)dt=\lim _{\epsilon \to 0^{+}}{\frac {1}{2\epsilon }}\int _{-\epsilon }^{\epsilon }\varphi (t)dt}$ which reduces to ${\displaystyle \int _{-\infty }^{\infty }\delta (t)\varphi (t)dt=\varphi (0)}$ whenever ${\displaystyle \varphi }$ is continuous at ${\displaystyle 0}$. 78.196.93.135 (talk) 00:25, 31 May 2017 (UTC)

I think perhaps it is your own reading comprehension that is a problem if you believe that people in this discussion think that the delta function is really a mathematical function, or if you think that the article gives this impression either. Also, the belief that this is the "most useful definition in Fourier analysis" is naive and wrong. I suspect it was copied from some textbook without understanding the wider mathematical context. Sławomir Biały (talk) 01:05, 31 May 2017 (UTC)
"Dirac delta function" is nothing but one of Oddities of mathematical terminology. Wikipedia mirrors the world. See also List of types of functions#More general objects still called functions. Boris Tsirelson (talk) 08:52, 31 May 2017 (UTC)
1. ^ Isham 1995, §6.2

## "The delta function only makes sense as a mathematical object when it appears inside an integral"

There has been a back and forth over whether the sentence in the lead "The delta function only makes sense as a mathematical object when it appears inside an integral" is a proper, accessible summary of the source, which says (Gel'fand & Shilov 1968, p. 1):

If the singular function occurs at all in the final result, it is only in an integrand where it is multiplied by some other sufficiently well-behaved function. There is therefore no actual necessity for answering the question of just what a singular function is per se; it is sufficient to know what is meant by the integral of a product of a singular function and a sufficiently 'good' function. For instance, rather than answer the question of what a delta function is, it is sufficient for our purposes to point out that for any sufficiently well-behaved function ${\displaystyle \phi (x)}$ we have ${\displaystyle \int _{-infty}^{\infty }\delta (x-x_{0})\phi (x)\,dx=\phi (x_{0}).}$

In addition, I have added a reference to Terry Gannon's article in the Princeton Companion, where he says in no uncertain terms

However, [the delta function] really only makes sense inside an integral.

Do we feel that the sources are being misrepresented in the context? If so, is there an equally accessible way to summarize the cited sources without that misrepresentation? Sławomir Biały (talk) 11:13, 27 July 2017 (UTC)

I note that an IP editor seems keen on having a discussion in article space. The purpose of these templates is not to engage in arguments in article space, but to indicate points that could be clarified or discussed, a request for clarification, not a mandate. Here is the long comment that I removed from the article:

on page 543 of The Princeton Companion to Mathematics Eq. 2 has a commutation relation equal a delta function , how is it that a delta function only makes sense inside an integral when this equation does not. Also, Green's functions are found by setting a differential equal to a Delta function without an integral. Third, Delta functions are use extensively in Quantum Mech. without being inside an integral. Isn't the state "makes sense" meaningless. What makes sense to me, might not make sense to you...

One has to consider the statement in the context of the article. We aren't talking about quantum mechanics or Green's functions. At the fundamental level to which such a statement is meant to apply, an equation like that of finding a Green's function, ${\displaystyle \Delta f=\delta }$, is merely shorthand for ${\displaystyle \int (\Delta f)\phi =\int \delta \phi }$ for all test functions ${\displaystyle \phi }$. I'm open to an equally accessible way to express this, but I think the current sentence is a fair summary for the target audience, which likely includes undergraduate engineers. For such readers, the sentence addresses the question of what the delta function "is". While pedants with a detailed knowledge of the subject may object to the niceties of the wording, they already know enough to not be misled by such pronouncements. Sławomir Biały (talk) 20:32, 27 July 2017 (UTC)

Many students are mislead by this statement and it hinders there ability to use delta function to solve physics problems. Wikipedia editors should at least make it clear that delta functions are (basically) never used "inside an integral" in practical problem solving (i.e. calculation). One student is convinced that the Kronecker delta "only make sense" when it appears inside a summation! — Preceding unsigned comment added by 131.252.127.172 (talk) 20:32, 29 July 2017 (UTC)

Ok. Would such students accept that the differential equation (for example) ${\displaystyle f''(x)=\delta (x)}$ is merely short-hand for the integral equation
${\displaystyle \int \varphi (x)f''(x)\,dx=\int \varphi (x)\delta (x)\,dx\qquad (=\varphi (0))}$
for all test functions ${\displaystyle \varphi }$? Sławomir Biały (talk) 21:09, 29 July 2017 (UTC)

Why do we insist on a test function? If "a vector" equal "b vector", then does not "a vector" dot "c vector" = "b vector" dot "c vector"? This test "c vector" (or function) at best just sits there through the calculation and at worst confuses even the smartest students.

For examples of how the Delta function is used in Quantum Mechanics:

"The Principles of Quantum Mechanics" P. A. M. Dirac Defines Dirac delta function and uses it to go from discrete basis to continuous basis. (Kronecker delta-> Dirac delta) p. 59

"Introduction to Quantum Mechanics" David J. Griffiths Uses Dirac delta to solve inhomogeneous Schroedinger eqn. by finding the Green's function. p.409

I see this disagreement in analogy with the ideas of imaginary numbers. At first people didn't think they made sense as mathematical objects. After all, x^2+1=0 has no root when you graph it. But later on when studying the cubic eqn. they saw the power of analysis using imaginary numbers. They could now find all three roots!

I have a feeling that most of the students reading this page are not Mathematicians rather Physicists and Engineers. These people want to know about the Dirac delta because they need to solve some problem that requires knowledge of it in their respected fields. They likely don't care about the mathematical niceties of distribution theory and whether it is Riemanan integrable or whether the limit exist etc. If they wanted to that know the niceties they would have taken Analysis courses. I gonna be bold and at the sametime very nonprecise in the next few statements. The Dirac delta maps a function onto itself. The Kronecker delta maps a vector onto itself. The Kronecker delta makes sense as mathematical object and so does the Dirac delta. Let's focus on what the Dirac delta is, not on what it is not. Let's focus on its power to solve some rather difficult problems without having to understand the details of Analysis and the theory of distributions.

131.252.127.172 — Preceding unsigned comment added by 131.252.127.172 (talk) 19:05, 30 July 2017 (UTC)

One doesn't need to understand the analytic details in order for the operational calculus to produce physically meaningful results. But to regard the delta function as a mathematical object requires test functions and integration. In the context of "a function that is equal to zero everywhere but the origin, where it is infinite, with total integral one", it is clearly important to point out that there is no such function, but the delta function can be made mathematically meaningful. This requires integrating against test functions. Sławomir Biały (talk) 21:03, 30 July 2017 (UTC)

First, you did not address the points I brought up in my last comment. Second, you did not explain why the Dirac delta cannot be made mathematically meaningful without a text function as you claim over and over. (And I repeatly ask you to explain.)

You are not even clear about what you take to be a mathematical meaningful object. (You just claim too much is being read into it) Does the identity element in a group have mathematical meaning to you? Does the identity matrix have mathematical meaning to you? In operational calculus the operator acting on the Green's function is equal to the delta function. Just like a group element and its inverse are equal to the identity element or the matrix and its inverse are equal to the identity matrix. Are we talking about some metaphysical ideas here or mathematics?

Why do you not back up your claims with clear/specific examples, like I do?

I can see you have dominated everyone in this talk page so you must feel very personal about what the Dirac delta means to you. Do you have any papers published where you use Dirac deltas? I am willing to learn... but let us not mislead students into believing that the Dirac delta does not have mathematical meaning.

131.252.127.172 — Preceding unsigned comment added by 131.252.127.172 (talk) 22:21, 30 July 2017 (UTC)

The article does not say that the delta function has no meaning. It says that "The delta function only makes sense as a mathematical object when it appears inside an integral." There is a lengthy quotation in a footnote, to Gelfand and Shilov: "There is therefore no actual necessity for answering the question of just what a singular function is per se." The parallel with the Kronecker delta is a false one. The Kronecker delta ${\displaystyle \delta _{ij}}$ is equal to one if ${\displaystyle i=j}$, and zero otherwise. But it is not the case that ${\displaystyle \delta (x)}$ is equal to zero for ${\displaystyle x\not =0}$ and equal to infinity for ${\displaystyle x=0}$: this is a meaningless statement. Sławomir Biały (talk) 23:09, 30 July 2017 (UTC)

You are so caught up on the fact the Dirac deltas are not functions that you will never see how useful (and powerful) they really are. And it seems like you are choosing to take young people down with you. :(

We already state on the page that the Dirac delta is not a function more times than I can count. Would taking the statement in question off the page really cause you so much discomfort that you are willing to argue to the bitter end to keep it up?

And I quote, "The delta function only makes sense as a mathematical object when it appears inside an integral."

            "In the context of "a function that is equal to zero everywhere but the origin, where it is infinite, with total integral one", it is clearly important to point out that there is no such function, but the delta function can be made mathematically meaningful."


I have given numerous examples of the mathematical meaning of Dirac delta outside the integral in my comments above. (You also have not answered the questions I ask you above.) There are many more in well accepted textbooks and published papers. You only have to look. Even your own cited books uses it outside an integral, and you still insist on being self righteous and use that quote to mislead people.

If you truly believe that Dirac deltas are not mathematical objects when they appear outside an integral, then you should submitted a paper to a peer reviewed journal where it can be judged by experts.

It is a shame a person like you has so much time and you choose to use it misleading young people on Wikipedia.

Don't bother replying if you are only gonna repeat that the "delta function as a mathematical object requires test functions and integration." It's getting tiresome. — Preceding unsigned comment added by 131.252.70.156 (talk) 23:59, 30 July 2017 (UTC)

### Comment and arbitrary break

Sławomir is correct. However, there is a formalism where the delta function can be handled without dragging along integrals that makes total sense (since it is once and for all rigorously established using integrals). There's a book in the GTM series,

written by a mathematician who throughout his career taught engineering and physics students (as well as students of pure mathematics), that highlights the soundness of delta function manipulation in the aforementioned formalism. Message: You can treat it engineering/physics style without worry. A mathematician should worry, it is his job. But for someone like Dirac (or bleaker copies or students of the non-mathematical variety), it would just be a total waste of time to worry. It isn't their job. YohanN7 (talk) 09:50, 31 July 2017 (UTC)

I'd add that just because the delta function needs to be regarded as a functional if we're talking about mathematics, doesn't mean that other attitudes about the delta function can't also be discussed. The overall perspective in such approaches is that one uses a particular realization of the delta function, say, as a limit of Gaussians (in quantum mechanics, esp) ${\displaystyle \delta (x)=\lim _{\sigma \to 0}\phi _{\sigma }(x)}$ where the limit is understood as "outside the integral". (I believe this is the attitude, for example, in the Griffiths textbook suggested above.) Very often such treatments will ask the student to "prove" identities regarding the delta function, where it is precisely this outside-the-integral-limit that allows for such a proof. So it really doesn't help students of physics and engineering either to pretend that the delta function can be treated mathematically independently of being thought of mathematically as an integrand or functional. This is not mere pedantry for its own sake: it is important for the basic mathematics to be correct. Sławomir Biały (talk) 12:47, 31 July 2017 (UTC)
No disagreement at all except maybe this: Had the basic mathematics been wrong, the formalism wouldn't have been around. Also, any responsible text or teacher will point out first thing that the delta function isn't really a function, at least not a function having all the advertised properties.
I guess the issue is this. Should the article
• Pretend that it is raining and don't mention formal use?
• Mention formal use and say it isn't rigorous?
• Mention formal use and say that the formal use can be backed up rigorously?
The first option ignores a big percentage of readers. The second option says (in effect) that most books are plain wrong. The third option is, to me, the verifiable (and IRL true) truth. I mean, even respected math texts use informal, but easily formalizable presentation. It the present case, we don't have "easily formalizable", but rather "with difficulty formalizable". The more reason so for allowing for it. YohanN7 (talk) 13:48, 31 July 2017 (UTC)
Please note, I haven't read the article in a while, just this talk thread. YohanN7 (talk) 13:17, 31 July 2017 (UTC)
I think an introductory section could be drafted illustrating one or two basic properties using the limit-outside-the-integral approach, and also the formal use in the applied setting. I think an example should be sufficient. Thoughts? Sławomir Biały (talk) 13:56, 31 July 2017 (UTC)
Sounds splendid. I'll see if I can find something "quotable" on the issue in that Vretblad book. YohanN7 (talk) 14:03, 31 July 2017 (UTC)
Something like
Regrettably, there is no ordinary real-(or complex)-valued function that satisfies these conditions. Condition 2 irrevocably implies that the integral in condition 3 must be zero. Nevertheless, using formal calculations involving the object δ, it is possible to arrive at results that are both physically meaningful and "correct"
is clearly citable or quotable verbatim. Since this issue is so "infected", it might be a good idea to have this quote (or some other like it) from an actual mathematician and not a physicist/engineer accompanying such an intro section. YohanN7 (talk) 14:51, 31 July 2017 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I was thinking of something that starts with an example. The simplest (?) example is giving an object a kick with a delta function force ${\displaystyle p\delta (t)}$ (note that this is dimensionally consistent). The object then goes from rest to uniform motion with momentum p. Model this by a sustained force ${\displaystyle F=p/\Delta t}$ over a time interval ${\displaystyle \Delta t}$, i.e.,

${\displaystyle F(t)={\begin{cases}p/\Delta t&0

Taking the limit as ${\displaystyle \Delta t\to 0}$ gives the force ${\displaystyle F(t)=p\delta (t)}$, whose affect can be felt only if we understand this limit as taking place only outside the integral. Possibly it could also be mentioned very early on that this shows that the delta functions should be regarded as the derivative of the Heaviside function, provided the limit is "outside the integral". Sławomir Biały (talk) 16:16, 31 July 2017 (UTC)

Looks reasonable as a basic example. The equation
${\displaystyle {\frac {dH(t)}{dt}}=\delta (t)}$
is definitely a part of the basic formal (it being implicitly understood that it should be used in a context) tool-kit. This necessary qualification (where and how to use it like your "limit outside integral" (does interpretation as distributional derivative work b t w?)) is usually left out, or at least not repeatedly mentioned in the engineering/phys literature. It is omissions of this kind that accounts for most of the non-rigor in everyday practical usage. Brief mention of this is perhaps something that could go in? YohanN7 (talk) 12:46, 1 August 2017 (UTC)
In examples, we shouldn't emphasize the fully rigorous interpretation, no. But it is only with the "outside-the-integral" view that sources in this subject are able to prove "theorems" about the behavior of the delta function, and thus develop a context for the validity of the operational calculus. Most students are willing to accept these formal rules for manipulating integrals, and so they also accept the operational calculus. For example, Bracewell declares as his definition of the delta function the limit ${\displaystyle \lim _{\tau \to 0}\int _{-\infty }^{\infty }\tau ^{-1}\Pi \left({\frac {x}{\tau }}\right)dx}$. It is defined likewise in Schaum's Outline of Quantum Mechanics (for example), and probably lots of places if one really looks. Various properties of the delta function then follow from elementary manipulations of the integrals like the mean value theorem for integration. So this is not a mere matter of implicit understanding: typically sources do say explicitly that the delta function is the limit of something. I'd much rather present it this way in an example than treating the operational calculus as a magical black box. The only mathematically deep problem is specifying in what sense this limit-outside-the-integral is actually properly a limit, and in what sense all of the different limits are the same: that requires distributions, but can safely be omitted from initial examples. Sławomir Biały (talk) 13:54, 1 August 2017 (UTC)
I agree 100%, and perhaps you have misunderstood me. The delta is introduced the way you describe (or similarly, detail and rigor varies). It is, of course, only after this introduction it is "understood" how to properly use formulas like ${\displaystyle {\frac {dH(t)}{dt}}=\delta (t)}$ and similar ones. I understand that a mathematician wouldn't write this formula, even if facing the business end of a shoot gun, but a physicist is never happier than when writing them. In practice, these formulas (alas) serve more like a "table of integrals" of sorts, than as instructions to do proper math (taking limits). I think this is quite sound since it saves time, and always gives the right answer if the practitioner knows what he is doing. (See also new sections below by IP. The bra-ket notation is even more riddled with problems of this sort (and of ambiguities), but the end result is magically always the right one.) YohanN7 (talk) 07:07, 3 August 2017 (UTC)

### Dirac delta in 3D

It might be helpful to students if we added the three dimensional versions of the Dirac delta in different coordinates (spherical, polar, etc.), for easy reference.

### Application to Quantum Mechanics

This section isn't using the standard position and momentum basis that beginning QM students would easily recognize (i.e. use p instead of y). Minor note, one of the equations is not in bra-ket form. Can we add a statement in this section to better show the ease of changing basis with Bra-ket notation like "bra x" "ket psi" = psi(x) and using identity operator to change basis. The use of the Dirac delta is standard in QM to do this. This would be similar to the history section with the Fourier analysis but with the compact Dirac notation that is used in the physics community.

## Is Dirac delta function a function?

Hi all. Because Dirac delta is a distribution, it automatically means that it is a real-valued linear functional, right? But being a real-valued linear functional makes it a real-valued linear map. The latter makes Dirac delta a real-valued map. And a real-valued map is a function. So long story short, Dirac delta is a function. However, the article explicitly says it's not. It's like saying that a field is not a set. Could anyone fix it please? Thank you. Konstantin Pavlovskii (talk) 12:21, 9 August 2017 (UTC)

Hello D.Lazard,

I've decided to talk to you because apparently you reverted my recent contribution. I am extremely new to the Wikipedia and I haven't had enough time yet to get used to how things work here.

You wrote in the edit summary of your revert that "The article was not wrong" and that "the edit summary is mathematically wrong". I assume that the latter refers to the edit summary of my contribution. I think it would have been a good idea if you had left a note on the article's Talk page explaining the reason why you think my edit summary "is mathematically wrong" before actually reverting my contribution, exactly as Wikipedia recommends.

Because the contribution is already reverted by you, I'd still -- even more so -- appreciate a note from you on the article's Talk page explaining the reason why you think my edit summary is "mathematically wrong". This way you'd give me an opportunity to respond to your concern, so we could reach an editing consensus.

Thank you, I hope to talk to you soon. Konstantin Pavlovskii (talk) 10:16, 10 August 2017 (UTC)

The Dirac delta function is not a function, as clearly stated in the article about it. This fact is sufficiently known and accepted by all mathematicians, for not needing any discussion here. Thus, this is your assertion in the edit summary that is wrong, not the result of your edit. I apologise for not having done the distinction in my own edit summary. Nevertheless the revert was justified, because it is always useful to recall that the Dirac function is not a function, despite its name. As the point is not the distinction between a function and a distribution, I have edited the sentence for less emphasizing on this point. D.Lazard (talk) 12:42, 10 August 2017 (UTC)
To D.Lazard: Hello D.Lazard, what makes you think that a Dirac delta function that is a distribution, is not at the same time a function? Any distribution is a linear functional, so any distribution is a real-valued map. By definition any real-valued map is a function. I've already pointed it out in the Dirac delta function talk page the corresponding article that you refer to has been recently edited by the community to remove the incrorrect statement that the Dirac delta function is not a function. You could have a look at the new Dirac delta article yourself. I'd appreciate if you undid the harmful reverts of my contributions, please. And could please discuss my contributions on the corresponding talk pages first before reverting them. And thanks for that prompt reponse. Konstantin Pavlovskii (talk) 13:34, 10 August 2017 (UTC)
I think Konstantin's perspective is reasonable. I don't claim to fully agree with it, but I'm inclined to tone down the "not a function" aspect at least. Russian mathematicians even call it a "generalized function" ;-) I think we should clarify both the sense in which it is and is not a function, rather than relying on possibly hidden meanings of the word "function". Sławomir Biały (talk) 13:43, 10 August 2017 (UTC)
A distribution is a linear functional, that is, a map from a space of functions to a space of functions to the reals. Thus Dirac delta function is not a real-valued function, although it may be considered as a "function-valued" function. As, for everybody, a function, without further specification, means a real- or complex-valued function, you are wrong by saying that Dirac delta function is a function. D.Lazard (talk) 13:10, 10 August 2017 (UTC)
D.Lazarad, a linear functional is always a map from a vector space to its underlying field. In this particular case, it is a map from a vector space of real-valued functions of a real variable. The latter is a vector space whose underlying field is reals. Therefore the distribution (linear functional) used to define the Dirac delta function is a real-valued linear functional. And once again, a distribution has to map a vector space to its own underlying field not any other field. Therefore the Dirac delta function is a distribution, which is a real-valued functional, and -- you already know -- it automatically makes Dirac delta function a real-valued map (e.g., a real-valued function). Konstantin Pavlovskii (talk) 18:13, 10 August 2017 (UTC)
OK, Dirac delta function is a map from a unspecified vector space of infinite dimension to the reals. Moreover, the domain of this map depends on the context (either the functions, in the usual sense, that are defined for x = 0, or the functions that are defined everywhere, or the continuous functions, etc). Thus a correct formulation would be: "although they are generally considered as a generalization of real-valued functions of one variable, distributions and Dirac delta function are not such functions". In all the articles you have edited, Dirac delta function appear in contexts (for example Fourier transform), which are generally devoted to real-valued functions of a real or complex variable. In such a context, "function" means function of a real or complex variable. It is thus important to recall to non-experts that Dirac delta function is not a function in this usual sense. Thus I would agree to replace "is not a function" by something like "is not a function in the usual sense". But I disagree to remove the caveats. D.Lazard (talk) 19:00, 10 August 2017 (UTC)
First of all I did not understand why you brought up the question of dimensionality of a vector space the DDF maps to real and what follows it -- all these other things you have there up to ". Thus a". I certainly miss the reason you wrote them. What's helpful is to make sure that the users, should they have any doubt about what the Dirac delta function really is, could in one click reach a page where the fact that the DDF does not take any real value anywhere on the real line -- is expicitly written and I would strongly recommend against using word patterns that support the weird idea that the DDF is a misnomer, for example: "DDF is not a function, in <certain sense>". An engineer perceives this as you if you were saying that DDF is only a function to a certain extent (and not to the full extent). Sort of like a peanut being not exactly a nut. This pattern will continue to generate and suppport that "misnomer" that I even saw on someone's personal page here. And also could I ask you D.Lazard again to revert the reverts of my contributions that you reverted and bring any concerns you have about the reader confusing DDF with a function of a real variable to the corresponding talk pages before reverting the contributions. I will check that there are no vague statements in the articles that would make domain of the DDF appear real. Finally, as for your text suggestion, "generally considered" sounds like a pretty toxic term to me, but as I am a new user, let me focus on the fact that your text suggestion is obviously not true in the first place: "although they are <..> considered <...> generalization of real-valued functions of one variable <...> Dirac delta function are not such functions". Well, DDF is a real valued function of an exactly one variable -- of a function. It's a linear functional -- hence the name. You might want to consider consulting the article in question:namely, the definitions section. And as a reminder, please, revert your reverts of my contributions related to the DDF and bring your concerns to the corresp. Talk pages. Thank you. Konstantin Pavlovskii (talk) 20:43, 10 August 2017 (UTC)
This is just a note that I replied at User talk:D.Lazard's user talk page, but it's fine by me if anyone to refactor this discussion so it's all in one place if they want. Sławomir Biały (talk) 13:44, 10 August 2017 (UTC)
Done. Konstantin Pavlovskii (talk) 09:19, 13 August 2017 (UTC)

Just a note. Anyone first rigorously learning the notion of a function in a course on set theory will be very open-minded as to what constitutes a function. So it is good that the article is clear on what is its notion of "function". YohanN7 (talk) 14:03, 10 August 2017 (UTC)

Every article using Dirac delta function or talking about it supposes that the reader has some knowledge of calculus. The the reference to set theory is not relevant here. If one does not clearly state that Dirac delta function, is not really a function, a reader of these articles will surely think that it is a real-valued function of a real variable (the basis of calculus), which it is not. Thus not distinguishing Dirac delta function from usual functions is misleading and confusing. Thus even if, in some cases, it may make sense to consider Dirac delta function as a function, this should be avoided in WP. Moreover, any edit that would imply that Dirac delta is a function should include a reliable source supporting this. D.Lazard (talk) 14:53, 10 August 2017 (UTC)
It is rather strange to read that the reference to the set theory definition is "irrelevant". It is akin to saying that vectors that aren't arrows in 3 aren't vectors (because they aren't encountered in Calculus 101). Note that all I was saying is that it is good that the article now is trying to be precise on what it means by "function". YohanN7 (talk) 08:49, 11 August 2017 (UTC)
D.Lazard, when the article says that the Diract delta function is a function it does at the same time explicitly say that it is not a function of a real variable, and I strongly suggest to add another clarification: "therefore it is not defined for any real argument". There is no source of confusion. However, just saying that "Dirac delta function is not a function" is not true. As simple as that. Saying that it's not a function of a real variable is true, but it begs a question: "Eeeh, but a function of what kind of a variable then? Or is it not a function in principle?". There is no way around making a statement about the functionhood of the Dirac delta (in terms of the concept of a function -- weird to even mention that). And the principal answer to this question can't be "no it's not a function" because it's not true. So to me it looks like there's no other option but to explain straightforwardly to the reader whatever is already in the article plus that additional clarification that I suggested in one form or another. And (perhaps I'm getting too emotional here) there has never been an issue of a "Dirac delta function" being a "misnomer", this issue was born on the pages of the Wikipedia and traces back to at least 2002 when the community for some reason let that (mind slip?) happen that suddenly gave birth to the problem of a "misnomer". It's a pretty well-known fact that any distribution is a function and any measure is a function. Why whould anyone even expect of something defined to be a distribution (or a measure) to suddenly not be a function? This is what would sound wild to anyone (and is not true, above all).Konstantin Pavlovskii (talk) 18:50, 10 August 2017 (UTC)
The Dirac delta function was not defined to be a distribution, though - it was defined to be a real-valued function which is zero everywhere except the origin and integrates to 1 on the real line. Of course this function does not exist. This article has a challenge, therefore, that it is using the name for a non-existing object (the Delta function) to refer to an existing object (the Delta distribution). That is what the phrase "the Dirac delta function is not a function" is meant to suggest. Whether the Delta distribution is "really" a function, under the hood, is not the point. The ambivalence in this article between the Delta function and the Delta distribution has increased over time; older versions such as [3] had their own flaws but were more clear in the lede about the function nature of the Delta function. Also see my comment at [4]. — Carl (CBM · talk) 19:40, 10 August 2017 (UTC)
Hi Carl. I am not getting it, sorry, you say: <<using the name for a non-existing object>>. There is no such thing as the name of an non-existing object. It's math, objects don't appear and then pass away. Any of them either exists or doesn't in general, they are not parts of material world. When an object is non-existing, so are all its properties including the name, definition, etc. It sounds even more confusing to me as I read further: <<That is what the phrase "the Dirac delta function is not a function" is meant to suggest.>> Suggest what? You can't check whether or not an object is a function if it doesn't even exist. This statement doesn't make sense in the absense of the object that it's talking about. Could you please clarify the rest, I think I'm stuck. Thanks. Konstantin Pavlovskii (talk) 23:05, 10 August 2017 (UTC)
"The smallest odd multiple of 8" is a name for a nonexistent natural number. Here's a long list of nonexistent objects at MathOverflow [5]. So there are many names in mathematics for nonexistent objects. The prototypical "Dirac delta function" is defined as a real valued function on the real line that is zero except at the origin and integrates to 1 on the real line. We can prove that no such real valued function exists, of course. When most people write "the Dirac delta function is not a function", they are simply alluding to this fact, to point out that they are using the term "Dirac delta function" to mean something else. For example, this answer at MathOverflow [6] shows the usage: "δ isn't a function - yet sometimes it behaves like one." — Carl (CBM · talk) 01:21, 11 August 2017 (UTC)
Oh, I see it was a joke. Jesus I thought you were serious :) Of course you can make up whatever name you want and pretend that it's a name for something that's proven to not exist under ZFC, say. But the latter statement (that it's its name) is... could you tell me (a stupid physicist) what's its status in ZFC? As for the Dirac delta function, I think I've said enough, and the historical reasons for the cognitive slip that you think made it grow into a <<fake "misnomer">>... This slip now looks way more dissapointing to me than before. It so hard to avoid those misfortunate congnitive malfunctions big and small... they grow through the years and a couple decades later their falsehood just becomes a part of you. And one day you realize you're not quite the person you've been trying to be, not at all. Konstantin Pavlovskii (talk) 02:40, 11 August 2017 (UTC)

Konstantin Pavlovskii takes formal definitions too seriously. They often mix an idea and its implementation. Often an idea has many implementations. For instance, some sources define a natural number to be a finite ordinal in such a way that it is true that ${\displaystyle 5\in 6,}$ but it is evil to say so ("evil" according to nLab, not to me). See "Equivalent definitions of mathematical structures". Likewise, it is evil to say that delta function is a distribution and therefore a function. After all, in ZFC everything is a set; but it is evil to say "5 is a set". This happens outside math, too; it would be evil to say "the fifth bit of this song is 0" even though this could be true according to a given mp3 file. It is not serious, to bee too much serious; it is rather evil. "Some subjects are so serious that one can only joke about them." (Niels Bohr) Boris Tsirelson (talk) 14:03, 11 August 2017 (UTC)

Hi Boris. The article clearly identifies a particular class of implementations of the Dirac delta function in the very beginning by saying that <<the Dirac delta function, or δ function, is a generalized function, or distribution on the real number line...>>. Seeing this why don't you say that the article itself already "mixes an idea and its implementation". And then the same article goes on to say <<...[is a distribution] on the real number line that is zero everywhere except at zero>>. Jesus one of two things: either this functional operates on a trivial vector space on R (i.e. the R itself -- just {0} doesn't work because in this case it wouldn't talk about functional values "everywhere except zero") or the author doesn't know what they are talking about. Perhaps the latter since the Dirac delta function defined in the "Definitions" section doesn't take any values on the real number line whatsoever. So, a quick summary:
1. The article as of now does mix an idea and its implementation in the top section.
2. The article as of now does talk about the values that "the Dirac delta function", a "distribution on the real number line" takes for real arguments (it doesn't take them in any of the implementations of the Dirac delta described in the article).
3. Then, a few lines below the article states that the values it was just talking about (see #2) don't exist by saying that <<The Dirac delta function is not a function <...> of real variables>>
If the article picks the "distribution" implementation in the first sentence, then what's wrong with sticking to it until the end of the paragraph at least? And let's fix these statements in the top section that contradict each other. Otherwise it's just not a sound article: it discredits the Wikipedia (the best case scenario) or fools the reader into believing them all at once (the worst case scenario). When I suggest how to fix them, my contribution gets reverted, even though it explains things straightforwardly to the reader sticking to the implementation introduced in the beginning of the section. Even saying <<in the usual sense of real variables>> is ambigous. What does it mean? If someone asked what's a "function in the usual sense of real varibles?" I would answer "a real-valued function of a simply ordered set of real variables". Does anyone disagree? Is it what's actually meant by this sentence? Konstantin Pavlovskii (talk) 04:23, 12 August 2017 (UTC)
Hi Konstantin. Yes, the article does mix an idea and its implementation, since this is usual in mathematical literature. Yes, another article "Distribution (mathematics)" treats distributions as linear functionals. But it never calls these functionals "functions" (in spite of the uncontroversial fact that they are functions); this is also usual in mathematical literature. Unless otherwise stated, a reader usually interpret "function" (in this context) as a function on R or Rn.
Now think about a text that defines a natural number to be a finite ordinal in such a way that it is true that in fact 5 belongs to 6. Unless devoted to implementation details, such text never says that 5 belongs to 6 (in spite of the uncontroversial fact that it belongs).
A reader of Wikipedia, being often not a mathematician, usually is not interested in implementation details.
Think also about an equivalent definition of the space of distributions as the completion of the space of test functions w.r.t an appropriate topology. Now implementation of completion matters; usually you get that a distribution is an equivalence class of Cauchy sequences (or filters, etc). Now a distribution is not a function!
Well, frankly I am not sure it is not; it depends on very subtle implementation details; apriori it may happen that a single set from the ZFC universe is both a function (from something to something) and an equivalence class (of something). But you surely feel that this is a stupid matter. Who bothers whether a finite sequence of bits can be a legitimate sound file in one operation system and a legitimate executable in another system? Well, maybe some expert in computer crime could... but surely not a typical user of a computer.
We do not hide implementation details from the reader, but we give them DUE WEIGHT, and not more. Boris Tsirelson (talk) 04:50, 12 August 2017 (UTC)
On your first paragraph: OK. You understand that "unless otherwise stated" bit at least. Because the question of the functionhood of the Dirac delta function is controversial enough (not in Russia though) to ignite fierce discussions and is a well-known public misconception, I suggest simply clearing up this question in the top section once and for all, at the same time clearly stating the nature of the DDF functionhood right in the same sentence or right next to it. If you prefer a "light version" of the same fix that's fine with me personally (I'd call this an editing consensus, although see the note below), then just stating that the DDF is not a function of a real variable is enough I think, it is exactly the change that was introduced right after I brought up this topic. I didn't have major concerns about the new way it was worded <<The DDF is not a function, <an addition about a real variable> >>. However, this light version is not extremely appealing to me as to a person who frequently has to clear the air in terms of whether the DDF is a function (fullstop). A first-year engineering student is very likely (in my experience) to interpret "The DDF is not a function, <blah-blah-blah> " as "The DDF is not a function, I understand this part, I don't care about what's after the comma". This is because students don't know much about the functions that are not functions of a real variable, so they take that little piece after the comma as an additional clarification of whatever is before the comma rather then a (very serious!) reduction of scope of the statement. That's why saying "The DDF is not a function of a real variable" is much better in my view.
About the ordinals, there's no wide misconception among the public (with serious consequences) on whether or not in the implementation they use 5 is an element of 6. It's not a practical matter, so mentioning this explicitly is not needed, certainly not in the top section of an article that talks about a set of natural numbers. But if in the impelementation you describe 5 is indeed an element of 6, then saying explictly that it's not is not an option, it would be like selling your soul to the devil. Because it's not true and you would be confusing the more advanced readers. So just not mentioning this fact (even though it might be true) is totally acceptable and is not confusing to anyone.
I don't insist on specifying the details to the reader. I would merely want the top section of the article to not contradict itself and to clear up the question of the DDF functionhood to those students/engineers/physicists who are exposed to the publicly accepted view of the DDF as of a misnomer.
About the alternative implementations of a distribution. The Wikipedia is not trying to write about everything in the world and certainly not at once. So, in my view, picking a particular implementation (of the DDF and of a distribution and of a natural number and ...) is neccessary in the quick description of the DDF in the top section. Unneccessary generalizations and "meta-language" of ideas is confusing enough even for experienced readers. I remember reading Canadian Criminal Code that has a list of definitions of what "a public nuisance" or "a public intoxication" (or smth like that I don't recall) mean in this particular section. It's not an option for an article in the Wikipedia to resemble a text of a legislation, it would scare off the readers. It is for a reason that ideas and implementations get mixed in the mathematical literature and in any literature. For example, talking about the Soviet past, a Russian text would say "the Communist party", however the implementations of a communist party are numerous. Speaking about a relatively complicated mathematical concept in the "meta-language" of ideas won't let you use the terminology appropriate to the common implementation of the concept.
So selecting a common implementation and sticking to it in the top section is a must. Whoever is interested in the alternative implementations are advanced enough to not get confused with a top section written like that, it does no harm to them, at the same time, less experienced readers benefit from it because this way it is more understandable and uses the terms that they are used to.
So the program that I suggest is:
a) (a must) fix the mutually contradicting and vague (like <<in the usual sense of real variables>>) statements in the top section while sticking to the common implementations of the mathematical concepts throughout the top section, then
b) clarify the issue of the functionhood of the DDF in the top section with straightforward and unambigious sentences that preferably don't use the "the DDF is not a function <comma>" word pattern, replacing it with "the DDF is not a function of a real variable". The latter being done with the sole intent of benefiting your average engineer or physicist who is not used to the fact that commas in math routinely and greatly reduce the scope of the statements. Konstantin Pavlovskii (talk) 06:46, 12 August 2017 (UTC)

I have edited the lead in order to avoid excessive formalism introduced by Konstantin.pavlovskii, and to keep the important functional property of Delta function. IMO, this new lead is much more informative for readers that have never heard of distributions, and, nevertheless, remains mathematically correct. D.Lazard (talk) 10:42, 12 August 2017 (UTC)

Excellent! Thank you Daniel and Sławomir. Now it looks like a decent top section to me. Konstantin Pavlovskii (talk) 16:09, 12 August 2017 (UTC)

## Softplus-inspired (smoothed) infinite delta function

I've found it to be so:

y = ln(2) / ( abs(x) - ln(cosh(x)) ) - 1

I like it, because it grows really nice and it spire is sharp as hell! Does this has a name?

And its finite modification:

1 - ( abs(x) - ln(cosh(x)) ) / ln(2) — Preceding unsigned comment added by Xakepp35 (talkcontribs) 01:36, 15 February 2018 (UTC)

That's not a model of the delta function. Sławomir Biały (talk) 02:07, 15 February 2018 (UTC)