Talk:Values of Dirac Delta Function over Reals

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Are we sure about this?

All the treatments of this subject are very careful not to state outright that $\map \delta 0 = \infty$, but instead skirt around the subject with the language of limits and such. The point is that $\infty$ is not a well-defined concept, as such.

I see it used on the Wikipedia page with the disclaimer "loosely" attached to it, (according to the references, possibly got from Gelfand and Shilov, where they discuss "generalised functions", difficult to get a definition of what they are except as "the sort of thing that the Dirac delta function is").

I find it defined this way in the Collins dictionary (Borowski and Borwein) but they have a tendency to be simplistic and populist.

Can we find a way of wording it so as to apply warnings to this? Unfortunately it's one of those results which everyone "wants" to be true, despite the fact that it actually may be "meaningless" in a rigorous mathematical sense. At base, I'm really not sure what to do about it. --prime mover (talk) 16:33, 15 March 2021 (UTC)

Great question. It certainly approaches $\infty$ in the limit. I see in Logarithm Tends to Negative Infinity, :$\ln x \to -\infty$ as $x \to 0^+$, so maybe here $\map \delta 0 \to \infty$ as $x \to 0$? That way, the site is internally consistent. --Robkahn131 (talk) 16:43, 15 March 2021 (UTC)
By Values of Dirac Delta Function over Reals
$\map \delta x := \begin {cases} \to \infty & : x = 0 \\ 0 & : x \ne 0 \end {cases}$
vs
By Values of Dirac Delta Function over Reals
$\map \delta x := \begin {cases} \infty & : x = 0 \\ 0 & : x \ne 0 \end {cases}$
Um no, that is not what it means at all, there is no "tending to infinity" there, $\map \delta x$ is a rock solid stone cold $0$ everywhere $x \ne 0$, there is no "tending to" infinity.
The best way to state it is as it is done in the definitions: it is the value of the height of the rectangle $\epsilon \times \dfrac 1 \epsilon$ (or $2 \epsilon \times \dfrac 1 {2 \epsilon}$ if you insist).
I believe that if it were possible to define $\map \delta 0 = \infty$ rigorously in this way, then the text books would do it that way. The fact is they don't, and they even tell us that we can't do that, by reminding us that $\delta$ is not actually a function.
As I say, what are you using as your source work? I am afraid that unsourced definitions are mistrusted on $\mathsf{Pr} \infty \mathsf{fWiki}$, we had too much of a contributor once thinking he or she was carving out vast swathes of brand new mathematics by inventing definitions which were at odds with the published canon. Hence we started a system whereby we strongly discouraged unsourced definitions by plastering the {{NoSources}} template all over them.
I believe this may be a case in point where the same may need to be done here. --prime mover (talk) 17:17, 15 March 2021 (UTC)
Sources
https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/step-and-delta-functions-integrals-and-generalized-derivatives/MIT18_03SCF11_s24_3text.pdf
http://hitoshi.berkeley.edu/221a/delta.pdf
https://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous%20Math/Delta%20Functions/Simplified%20Dirac%20Delta.pdf
https://www.iist.ac.in/sites/default/files/people/IN08026/delta_function.pdf
https://www.sciencedirect.com/topics/engineering/dirac-delta-function
https://en.wikipedia.org/wiki/Dirac_delta_function
https://www.thoughtco.com/dirac-delta-function-3862240
I agree - not a function -
I agree - $\map \delta x = 0$ if $x \ne 0$
I agree - $\map \delta x \to \infty$ not accurate
I still think the definition in place is fine as is
Thoughts? --Robkahn131 (talk) 18:32, 15 March 2021 (UTC)
Without intention to sound arrogant against certain fields, I note that many of these are works in engineering, where this might be an acceptable abuse of notation. I think it would be much more in keeping with the spirit of the site to afford $\delta$ a proper formulation as a distribution. Caliburn (talk) 20:54, 15 March 2021 (UTC)
I'm not sure I trust those sources which propagate this fiction. It's distressing to find that a fact that was carefully educated into me by people who clearly knew what they were doing being so casually dismissed now in numerous online papers, with barely an attempt even to explain what $\infty$ actually means in this context. And what does it mean? $\aleph_0$? $\aleph_1$? The "point at infinity"? It is only a convenient fiction, used to give an intuitive understanding to the concept.
Then we ask: is this a definition or a result? The derivation is problematic, and I don't believe it's a valid technique to use $\dfrac 1 {2 \times 0}$, which is undefined at any level beyond high school. As it is, you seem to have calculated $\dfrac \infty 2$, which would seem to lead to $\ds \int_{-a}^a \map \delta x \rd x = \dfrac 1 2$ (that's why I have little time for that added complication of trying to force it to be an even function, it adds confusion). Hence the derivation is handwavey at best, and does not actually convince the reader that the axioms of mathematics are fulfilled.
At base, while you can sort-of say $\map \delta 0 = \infty$, you can't actually use that knowledge. The only time you can use the delta function is in the context of the integral, where $\ds \int_\R \map f x \map \delta {x - a} \rd x$ is used to "sample" the function $f$ at the point $a$. There is really not a great deal else you can do with it.
When you define it using the $\epsilon$ technique, it's defined as the limiting value of some number $\eta$ such that as $\epsilon \to 0$ then $\epsilon \times \eta = 1$.
All I can do is repeat the received wisdom of whatever maths courses I did, and scan through all the books I can find on my shelf that go into the details of exactly what it means to say that $\map \delta 0 = \infty$. Some are robust in their refusal to admit the possibility that it may mean anything at all (e.g. 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.)), while some are casual about it (e.g. 1986: Kurt Arbenz and Alfred Wohlhauser: Advanced Mathematics for Practicing Engineers) and others don't even bring the subject up except to say that $\delta$ is "not a function".
TL;DR: I need to go away and ask someone who knows what they are talking about on this subject. But if it were up to me, I would relegate the results on this page to another of those "footnote" sections (bear with me, I have absolutely no ability to communicate this evening, I can't even put my fingers on the stupid keys without going back to correct half of my keystrokes). --prime mover (talk) 21:25, 15 March 2021 (UTC)

Some feedback from MathStackExchange

I have had a fair few responses to this, which I will transcribe here:

No. Defining $\delta(0)=\infty$ is not at all right, because it has to equal $\infty$ in just the "right way". You can define this as a distribution, which ignoring all technicalities, just means that it is an "evaluation linear map". i.e given an appropriate vector space $V$ of functions $\Bbb{R}^n\to\Bbb{C}$, we consider $\delta_a:V\to\Bbb{C}$ defined as $\delta_a(f):= f(a)$. In other words, $\delta_a$ is an object which eats a function as input and spits out the value of the function a the point $a$.
Afterall, this is literally what we want it to do: when we write $\int_{\Bbb{R}^n}\delta_a(x)f(x)=\int_{\Bbb{R}^n}\delta(x-a)f(x)=f(a)$, we literally mean that $\delta_a$ is an object such that whenever we plug in a function $f$, we get out its value at $a$. Of course, writing it in this way inside an integral is apriori just nonsense, because there is no function $\delta_a:\Bbb{R}^n\to\Bbb{C}$ for which the above equality can hold true.
So, in summary, the dirac delta is a function, but it's just that the domain of the dirac delta is a space $V$ of functions, and the target space for $\delta_a$ is $\Bbb{C}$. In short, it is the "evaluation at $a$ map".
As a side remark: the dirac delta is not in any way magical/esoterical once you think of it as an evaluation map. The concept of a function as being a mapping from one set to another set is (from our luxorious perspective of having hindsight) a completely standard concept. So, in this regard the dirac delta is simply a function/mapping. THe only difference is that the domain is a space of functions.
Furthermore, the concept of evaluation maps is a very basic concept in linear algebra (eg if you study the isomorphism between a finite-dimensional vector space and its double dual you'll see exactly what I'm talking about).
Now, the "difficulty" which comes with this is the question of "how to do calculus with these new types of objects". What I mean is in the ordinary setting of discussing functions $f:\Bbb{R}\to\Bbb{R}$, we have a notion of convergence/limits (i.e a topology), we have a notion of derivative (the study of differential calculus), and we have the notion of anti-derivatives/finding primitives etc. Extending these ideas to the more general setting is where the heart of the matter lies, and to fully appreciate that one should study more closely Laurent Schwartz's theory of distributions.
peek-a-boo (https://math.stackexchange.com/users/568204/peek-a-boo), Is it meaningful to define the Dirac delta function as infinity at zero?, URL (version: 2021-03-15): https://math.stackexchange.com/q/4063290


The $\delta$ "function" is not really a function at all. We define $\delta$ as a sort of special notation. When we write
$\int \delta(x) f(x) dx$
This is just "syntactic sugar" (shorthand) for $f(0)$. It has many of the same properties that an ordinary function does - for example,
$\int \delta(x) (f(x) + g(x)) dx = \int \delta(x) f(x) dx + \int \delta(y) g(y) dy$
But do not mistake $\delta$ for a "real function". It is not one.
Mark Saving (https://math.stackexchange.com/users/798694/mark-saving), Is it meaningful to define the Dirac delta function as infinity at zero?, URL (version: 2021-03-15): https://math.stackexchange.com/q/4063286


Adding to the existing answers and comments, I think a good way to argue against the slogan "$\delta(0)=\infty$" is to point out how limiting it is with respect to developing intuition for *other things of the same "type" as $\delta$ itself*. The $\delta$-"function" is, as others have said, not a function; rather, it's just something which can be integrated, which is not quite the same thing!
Making this rigorous leads to a lot of very interesting mathematics, to which the whole "$\delta(0)=\infty$" slogan is (in my opinion at least) a conceptual obstacle. Even if it doesn't get in the way of how one works with $\delta$ specifically, ignoring subtleties early on will just make things harder when they become central later.
Noah Schweber (https://math.stackexchange.com/users/28111/noah-schweber), Is it meaningful to define the Dirac delta function as infinity at zero?, URL (version: 2021-03-15): https://math.stackexchange.com/q/4063295

Summary

I would like us to lose this page.

If we must put this in somewhere, my view would be in an "also defined as" with caveats that "This is what (sneer) engineers do: $\map \delta x := \begin {cases} \infty & : x = 0 \\ 0 & : x \ne 0 \end {cases}$, but be aware that $\map \delta 0 = \infty$ is limited in meaning and usefulness, and gets in the way of understanding what it really means."

Would you be happy with that? --prime mover (talk) 22:21, 15 March 2021 (UTC)

I have learned a great deal here. I greatly appreciate the time you spent looking into the matter. I am completely on board with deletion. --Robkahn131 (talk) 22:59, 15 March 2021 (UTC)
I'll leave it up overnight, at least, so that other people can see what has been going on in the last few hours.
If we want to really flesh out understanding, I recommend that the subsidiary definitions as appearing in the "Also see" section under the refactoring flag of Definition:Dirac Delta Function all be given their own page, and perhaps, considering the subject matter, going against the usual $\mathsf{Pr} \infty \mathsf{fWiki}$ standard style and generating a page-with-subpages "Further Definitions" or whatever. I believe graphs would be a good thing to use here. --prime mover (talk) 23:13, 15 March 2021 (UTC)
The last chapter of Sasane is about the theory of distributions, and Dirac distribution is defined there. Actually, there is even an exercise asking to prove that "Dirac function" does not exist. I could try to extract definitions and the easiest examples. Unless somebody else is already looking into a more authoritative source.--Julius (talk) 17:30, 16 March 2021 (UTC)
I don't know what an "authoritative" source is, because there's always someone going to complain it's not rigorous enough.
I used the definition from Spiegel's Laplace Transforms. I don't know where Robkahn131 got his from, but he claimed his source trumped mine because his uses the one defined using the symmetrically placed rectangle function. I never got round to working through its definition in any of my other books because it's hard work ploughing through all the preliminary stuff before I get to it, and usually Delta function is later on in any of the books than I have been able to work through. --prime mover (talk) 19:26, 16 March 2021 (UTC)
Page 2 of the reed.edu link above says: "...some mathematically well-bred people did from the outset take strong exception to the δ-function. In the vanguard of this group was John von Neumann, who dismissed the δ-function as a “fiction,” and wrote his monumental Mathematische Grundlagen der Quantenmechanik2 largely to demonstrate that quantum mechanics can (with sufficient effort!) be formulated in such a way as to make no reference to such a fiction. The situation changed, however, in , when Laurent Schwartz published the first volume of his demanding multi-volume Th ́eorie des distributions. Schwartz’ accomplishment was to show that δ-functions are (not “functions,” either proper or “improper,” but) mathematical objects of a fundamentally new type—“distributions,” that live always in the shade of an implied integral sign. This was comforting news for the physicists who had by then been contentedly using δ-functions for thirty years. But it was news without major consequence, for Schwartz’ work remained inaccessible to all but the most determined of mathematical physicists."
Neumann and Schwartz are both waaaaaay beyond me and they didn't seem to agree. I see value in posting the best material that we can and add warnings and caveats that it is not a settled matter. Sweeping it under the rug seems to have less value. --Robkahn131 (talk) 19:45, 16 March 2021 (UTC)
Again, if you wish, I can write up a definition for Dirac distribution not using epsilons or infinities. Instead, it involves test functions, compact support and some basic integral properties. Then, as long as your triangular, rectangular or any other example satisfies those conditions, we are done. But these explicit infinities or limits to infinities will have to go because they carry no information about how actually we are getting to this infinity.--Julius (talk) 20:52, 16 March 2021 (UTC)
It's definitely worth having a go. If you have a number of different rigorous ways to define it, and they can be "named" distinctly, then Definition:Dirac Delta Function/Definition using Rectangle Function, /Definition using Test Function, /Definition using Compact Support, etc. etc.
Test functions and compact support will be used in the same definition. Now the question of equivalence of definitions will have to be reformulated. Instead what we will have is a list of properties a candidate "function" should satisfy. Then everything you see at the bottom here will require a proof that they fit in the Dirac bucket.--Julius (talk) 22:23, 16 March 2021 (UTC)
Whatever sense you can make of it. Seriously appreciate the time you can take away from whatever else you have scheduled for yourself. --prime mover (talk) 22:57, 16 March 2021 (UTC)
Then we can make an attempt to clear up the Augean stables by setting up a page / suite of pages / category discussing various popular non-rigorous approaches and explaining why they are non-rigorous and therefore "unacceptable" approaches.
For all that we are trying to branch out into, and support the frameworks for, applied mathematics, physics, statistics, economics, etc., all the fun stuff, at base we are rooted more in pure maths and foundational aspects rather than just being a here-are-the-formulae, go-away-and-plug-in-the-numbers sort of site. And sorry, but the clouds of dust that this excursion has kicked up is daunting me to the extent that I can't see the floor any more. --prime mover (talk) 21:02, 16 March 2021 (UTC)