# Talk:Differintegral

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## Oddity

Moved here from initialized fractional calculus;

A certain oddity about the differintegral should be pointed out. If the differintegral is "uninitialized", then although:

${\displaystyle \mathbb {D} ^{q}\mathbb {D} ^{-q}=\mathbb {I} }$

(That is, Dq is the left inverse of D-q.), the converse is not necessarily true.

${\displaystyle \mathbb {D} ^{-q}\mathbb {D} ^{q}\neq \mathbb {I} }$

But hold on, the ship isn't sunk yet! Let's take a look at integral calculus, to get a better idea of what's happening. First, let's integrate, then differentiate, using the arbitrary function 3x2+1:

${\displaystyle d\left[\int (3x^{2}+1)dx\right]/dx=d[x^{3}+x+c]/dx=3x^{2}+1}$

Well, that was pretty straightforward, and it worked. Now, what happens when we exchange the order of composition?

${\displaystyle \int [d(3x^{2}+1)/dx]dx=\int 6xdx=3x^{2}+c}$

Hmmm... I think it's fairly obvious what that integration constant is. Even if it wasn't obvious, we would simply use the initialization terms such as f'(0) = c, f' '(0)= d, etc. If we neglected those initilization terms, the last equation would fail our test. This is exactly the problem that we encountered with the differintegral. If the differintegral is initialized properly, then the composition holds. The problem is that in differentation, we lose state information, as we lost the c in the first equation. (see dynamical systems).

In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complimentary function is needed, not just a constant or set of constants.

${\displaystyle {}_{a}\mathbb {D} _{t}^{q}f(x)={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dx^{n}}}\int _{a}^{t}(x-\tau )^{n-q-1}f(\tau )d\tau +\Psi (x)}$

Charles Matthews 06:41, 22 Apr 2004 (UTC)

## oddity.. almost duplicate..

Moved here from initialization of the differintegrals (NB this duplicates the above, but not exactly):

Before discussing initialization of the differintegrals in fractional calculus, a certain oddity about the differintegral should be pointed out. Although:

${\displaystyle \mathbb {D} ^{q}\mathbb {D} ^{-q}=\mathbb {I} }$

That is, Dq is the left inverse of D-q. On first glance, the converse is not necessarily true.

${\displaystyle \mathbb {D} ^{-q}\mathbb {D} ^{q}\neq \mathbb {I} }$

However, let's take a look at integral calculus, to get a better idea of what's happening. First, let's integrate, then differentiate, using the arbitrary function 3x2+1:

${\displaystyle d\left[\int (3x^{2}+1)dx\right]/dx=d[x^{3}+x+c]/dx=3x^{2}+1}$

The process did work successfully. On exchanging the order of composition:

${\displaystyle \int [d(3x^{2}+1)/dx]dx=\int 6xdx=3x^{2}+c}$

The integration constant here is clear. Even if it wasn't obvious, we would simply use the initialization terms such as f'(0) = c, f' '(0)= d, ect. If we neglected those initialization terms, the last equation would fail our test.

This is exactly the problem that we encountered with the differintegral. If the differintegral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, we lose information, as we lost the c in the first equation. (see dynamical systems).

In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complementary function is needed, not just a constant or set of constants. We call this complementary function "${\displaystyle \Psi }$".

${\displaystyle \mathbb {D} _{t}^{q}f(x)={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dx^{n}}}\int _{0}^{t}(x-\tau )^{n-q-1}f(\tau )d\tau +\Psi (x)}$

(Working with a properly initialized differintegral is the subject of initialized fractional calculus).

Charles Matthews 06:44, 22 Apr 2004 (UTC)

Re the above - I have not managed to understand any exact statement on 'initialization', here. I think both of the above are too tentative to stand on a page for ever. The initialization issue, I think, is to do with what I wrote on fractional calculus about needing to take into account boundary conditions, because the fractional operators are not in general locally defined. I have had serious doubts about whether the Weyl differintegral (periodic boundary conditions) is really 'the same as' the others.

Charles Matthews 07:29, 22 Apr 2004 (UTC)

Yes, the boundary conditions and initialization amount to the same thing. Initialized fractional calculus refers to a paper published by the NASA John Glenn Research Center, cited on the main page. The problem with merely specifying the 'boundary conditions', in the sense of a function integrated from 'a' to 'b', or even specifying f'=w, f' '=x, f' ' '=y, etc, is that this does not fully specify the region of integration. One needs an infinite set of such constants: an entire complimentary function.
Regarding the Weyl differeintegral: perhaps it has a set of applications unique to it, but this does not at all preclude it's equivalency in regards to fractional calculus. In all the literature I've seen on fractional calculus, the Weyl differintegral has been treated without prejudice, although it is not used as often as, say, the Riemann-Louiville differintegral. If not ininitalized properly, the composition rule does not hold on either of them, and they produce different results. But when initialization is taken into account, they become indistinguishable from each other in effect.
However, I am not well-versed on the Weyl differintegral, and I'm not familiar with the application that you have studied it in. Even so, a difference in application does not imply a difference in fundamental formal properties, and as long as your argument rests on that leap alone, I consider it unsubstantiated.
Honestly, I could care less about the Weyl differintegral. I'm just telling you my experience and logic. It would be prudent for you to check out the literature before making any criticisms or exclusions. Kevin Baas 23:39, 22 Apr 2004 (UTC)

## NASA citation

Please tell me if the NASA paper you cite has been published in a refereed journal.

The Weyl version, as I have said before, is something referred to in a major treatise, Zygmund's Trigonometric Series. It should therefore be given a proper write-up. The whole point, according to G. H. Hardy, of doing mathematical analysis, is to get beyond fundamental formal properties.

Charles Matthews 08:17, 23 Apr 2004 (UTC)

NASA:I do not know where it has been published. I do know, however, that it is mathematically sound, and that it is applied. But I was not complete: The complementary function goes back to the early beginnings of fractional calculus: it goes back to Riemann. Some of the books cited on the mother page give reference to this.
Weyl: I don't dispute the fact that it should be given a proper write-up. Whatever gave you this impression?
The point: the point is not "what is the point of mathematics?" - that is off-track. The point is that a square and a triangle are both geometric figures. Likewise, a Weyl and a Riemann-Louiville differintegral are both differintegrals. Everything discussed in the fractional calculus section applies equally well to the Weyl differintegral as any other differintegral.
It is certainly possible to have unique content on the Weyl differintegral page, and to have links to and/or from topics regarding trigonometric series. This is not precluded by the weyl differintegral being linked from the fractional calculus discussion, as well as discussed therein. Wikipedia is not heriarchial. Kevin Baas 17:04, 23 Apr 2004 (UTC)

If this integral calculus satisfies the rule of Weyl differintegral defined on Wiki, can Wyel differintegral be applied to the integral calculus? If one integral differential operator is composed of an integral differential operator depend on space (x,y,z) and a quantity depend on space (x,y,z), can we consider it as some kind of fractional calculus? —Preceding unsigned comment added by Wadewizard (talkcontribs) 07:13, 3 August 2010 (UTC)

Weyl differintegral, and all other differintegrals for that matter, are equivalent to standard integer-order integral calculus when the order of integration/differintegration is an integer. Any mathematics that involves integration and/or differentation to non-integer orders can be considered as some kind of fractional calculus. Any that do not, or cannot be shown to be equivalent to (or where the change in a single variable or constant does not lead to a proportional change in the order of integration (i.e. the equivalence is only superficial)), are not fractional calculus. Integration and/or differentation to non-integer orders is any thing that meets the criteria for the fractionalization of operators, and specifically the integration / differentation operators. And most importantly / notably, a) the sub-class rule alluded to above (equivalent to integer-order version when the order is integer.), and b) composition ( f(f(a, order q), order p) = f(a,order q+p)). Kevin Baastalk 17:01, 4 August 2010 (UTC)

## ImportantLabeledEquation

I don't quite like the dotted boxes, be they made with blockquote or with the new ImportantLabeledEquation template. I will post this as discussion on Wikipedia talk:WikiProject Mathematics. Oleg Alexandrov 3 July 2005 01:06 (UTC)

## Definitions and explanation

This page needs a definition of the differintegral at the top of the page. All the equations need to define what all the variables inside it mean. The article on fractional calculus doesn't give decent enough perspective for someone to understand this page. Like.. what do the superscripts mean? Fresheneesz 17:22, 26 November 2005 (UTC)

The notation is an extension of the notation for differential. Understanding calculus is a prerequisite for understanding fractional calculus. If you understand calculus, you're familiar w/the notation, thus if you're not familiar w/the notation, you don't understand calculus, and thus you won't be able to understand fractional calculus regardless of whether or not we explain the notation. Kevin Baastalk 14:44, 17 June 2009 (UTC)

## relationship among definitions

This article should explain the relationship between the various definitions it gives. Are they all equivalent? If not, which ones differ, and how? Do they all have all the properties listed further down?

Also, what's the basis for the distinction between "standard" definitions and definitions via transform? The definition via the Fourier transform appears the most obvious to me; why is it less standard? Joriki 23:43, 13 February 2006 (UTC)

I don't know where the term "standard" comes from - i see your point here - but i don't know what we'd name that section otherwise.
As to the different definitions, I would say they differ in two basic ways: which ones work best in different application for solving a problem (for instance, the grunwald is more difficult to work with, but more general than the riemann one), and how the act under composition (for instance, transformed ones should hold fine under composition, but the "standard" ones need special attention which ultimately make them less general). Kevin Baastalk 14:49, 17 June 2009 (UTC)
I added a bit of explanation to this extent in the grunwald-leitnikov section. Kevin Baastalk 15:50, 17 June 2009 (UTC)

## Composition rule

Does the composition rule hold always without exception? Consider this:

${\displaystyle \mathbb {D} ^{\frac {5}{6}}\mathbb {D} ^{2}(t)=\mathbb {D} ^{\frac {17}{6}}(t)={\frac {1}{\Gamma (-{\frac {5}{6}})t^{\frac {11}{6}}}}}$

However, it is clear that

${\displaystyle \mathbb {D} ^{2}(t)=0}$--

Therefore

${\displaystyle \mathbb {D} ^{\frac {5}{6}}\mathbb {D} ^{2}(t)=0}$

--88.101.219.136 08:26, 4 July 2007 (UTC)

Yeah, there are some issues like that with composition. It's discussed to some extent in the literature, but I don't think it's made clear in the article. Kevin Baastalk 15:10, 9 June 2009 (UTC)
I tried explaining here. To give a short answer to this particular example: when you differentiated to the 5/6th order, you were integrating. And when you integrate, you've got to remember to add the " + C". The literature might not always have that taken into account and the equations on here might thus not be accurate in that respect (verifiability, not truth) - but that's what's happening. I went into some depth about " + C" and composition, esp. in relation to fractional calculus in the link I just provided. If you can understand it (i think it needs some work), it might help answer your question in the general case. Kevin Baastalk 16:33, 9 June 2009 (UTC)

## Error in the definition via Fourier Transform?

The definition via Fourier transform gives the Fourier Transform as a derivative as a product of i*t times the original function but the multiplication takes place in the frequency domain and not the time domain, so it should be i*omega. I am making the appropriate change. —Preceding unsigned comment added by 152.2.176.132 (talk) 15:06, 17 October 2007 (UTC)

I think that the definition via Laplace transform is incorrect as well. The correct rule for differentiation is
${\displaystyle {\mathcal {L}}\left\{{\frac {df}{dt}}\right\}=s\cdot {\mathcal {L}}\left\{f(t)\right\}-f(0)}$
--195.113.191.162 (talk) 07:33, 14 December 2007 (UTC)
Note that the "-f(0)" comes from integrating from 0 to x, but we are not always integrating from 0. There are different methods of differintegrating, not all of which preserve composition nicely, so the "-f(0)" isn't accurate for differintegration, and the more general
${\displaystyle {\mathcal {L}}\left\{{\frac {df}{dt}}\right\}=s\cdot {\mathcal {L}}\left\{f(t)\right\}}$
is preferable. Also note that since L is a transform, it represents the function as a complete orthogonal system, thus, so long as the coefficients on the (continuous) orthogonal basis (in this case, "s") are not evaluated (and thus none get multiplied or divided by zero), all information about the function is preserved, meaning composition holds w/out the "-f(0)". This implies that the "-f(0)" is, in fact, wrong. Kevin Baastalk 14:16, 17 June 2009 (UTC)

## Merger? and applications

First, this is at least 50% a duplicate of the more fundamental article on fractional calculus. Since it is also a "low-priority" article, perhaps it should be merged?

Second, on neither of these pages is there any discussion of application. I understand that it has application at least in analyzing so-called "meta-materials" (e.g. optical media with negative refractive index) - it would be awesome if I could read about this here; I've only heard hints of it from those in the know. 208.120.110.46 (talk) 03:14, 7 April 2008 (UTC)

A question has been raised concerning our ability to link to http://djvu.504.com1.ru:8019/WWW/c9a77d725039a74f68f7631d14c82f7d.djvu. I have removed the link pending clarification of licensing, [1]. I will raise the matter at WT:EL to help clarify these concerns. Please help resolve this issue by noting any evidence that material hosted is licensed for display and that a direct link to it is not a violation of WP:LINKVIO. --Moonriddengirl (talk) 18:19, 16 June 2009 (UTC)

So raised. I've also asked for feedback at WP:COPYCLEAN. --Moonriddengirl (talk) 18:24, 16 June 2009 (UTC)
A suggestion: Next time give the reason why the link is disputed so people don't have to track it down.
Plus, didn't Google get sued and subsequently pay millions in a settlement? I think that including a link to a copyrighted work hoping that it's not a problem is a bad idea. – Toon(talk) 21:34, 16 June 2009 (UTC)
I don't think Google Books is even a good comparison. The main page for this russian book search is at http://www.poiskknig.ru/. They explain that they do not host the copies and maintain they have no liability; their site is just an index of electronic copies of books hosted elsewhere. There's certainly a good possibility that many of the link files violate someone's copyright. This doesn't sound to me like a site we should be linking to at all. Shell babelfish 21:50, 16 June 2009 (UTC)
It looks straightforward to me: once the matter has been questioned the onus should be on the person wishing to include it to show evidence that the work is licensed, and in the absence of such evidence the link must go. There are several reasons why Google Books is irrelevant: we need evidence that this file is legitimately licensed. JamesBWatson (talk) 12:11, 17 June 2009 (UTC)

## Which is clearer?

Which of the following is clearer?

Option 1:

In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus. The operator does not define a separate function, but is a notation style for taking both the fractional derivative and the fractional integral of the same expression. This operator is here denoted

${\displaystyle \mathbb {D} _{t}^{q}.}$

See the page on fractional calculus for the general context.

Option 2:

In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by

${\displaystyle \mathbb {D} ^{q}f}$

is the fractional derivative (if q>0) or fractional integral (if q<0). In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.

Personally, I find the latter version preferable. (What does the phrase "does not define a separate function" mean, for example?) If there is a dispute about this, then I suggest that it be escalated by placing an {{expert}} tag on the article and an appropriately neutral thread at WT:WPM, rather than reverting good-faith edits to the article. Thanks, Le Docteur (talk) 00:51, 24 October 2009 (UTC)

Option 2 is obviously clearer. Kevin Baastalk 15:11, 17 March 2010 (UTC)

## spatial relations and composition of differintegration w/orthogonal spaces

you'll notice that if you integrate an n dimensional function (i mean orthogonal dimensions i.e. x,y,z) over an m dimensional region, it will be an m-th order integration that results in an n-m dimensional function.

I(F(x{n)),R{x{m}}) = f(x{n-m})

as differentiation is the opposite of integration, the reverse should also hold. that is:

d(f(x{n-m)),R{x{m}}) = F(x{n})

where the d is an m-th order differentiation. and you'll notice it requires a region of differentiation. this "region of differentiation" (my own wording, i know) is the "differentiation with respect to". I'm speaking kind of roughly here, but my point is that you have to consider that:

1. mth order integration of an n-dimensional function over an m-dimensional region results in an n-m dimensional function (namely, the dimensions in the integrated function but not the region)
2. and so the inverse operation, differentiation, necessarily has the reverse relations, if composition is to hold.
3. i.e. if going one direction requires a region of integration, then going the other direction requires something like the reverse.
4. when applying this to fractional order orthogonal integrations/differentiations, the region of integration/differentiation is necessarily of fractional dimension (fractal) and thus so is the result and/or the function being integrated/differentiated.
5. and furthermore there is the issue i discussed (again, roughly) here.

of all the literature i've read the closest that comes to addressing these mathematical issues is a paper by a division of NASA called "initialized fractional calculus". yet neither it, or any of these issues, are mentioned anywhere in the article. Kevin Baastalk 15:34, 17 March 2010 (UTC)

## chain rule?

can someone put the chain rule of differintegration in the basic formal properties section?

Kevin Baastalk 19:47, 2 June 2010 (UTC)

I just wanted to check to see if the ${\displaystyle n}$ in

 ${\displaystyle ={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{a}^{t}(t-\tau )^{n-q-1}f(\tau )d\tau }$

is an integer, designating the integer order and ${\displaystyle q}$ is the fractional order?

If so shouldn't this be rewritten to be

 ${\displaystyle {}_{a}\mathbb {D} _{t}^{q-n}f(t)}$ ${\displaystyle ={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{a}^{t}(t-\tau )^{n-q-1}f(\tau )d\tau }$ ?

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