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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)).
2004:. 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.
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2364:-- this is not true. If we determine that links to Google books are used to try to get around copyright laws then they are not allowed. If there are links that violate that they should be removed, not used as rationale to keep other copyright violation links. The existence of spam links on some articles doesn't mean spam is allowed either. If this link is allowed then we might as well not have any prohibitions against copyright violations, and that will never happen.
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1660:. 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.
2405:. 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.
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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
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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
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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
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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
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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. Knowledge is
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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
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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
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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
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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
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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
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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?
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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
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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,
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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.
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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.
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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.
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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
2279:(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.
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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
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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
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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.
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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.
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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?
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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?
2468:. 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
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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
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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)
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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
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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
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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
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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?
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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
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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.
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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
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1851:{\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}}}}}
786:{\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)}
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transformed ones should hold fine under composition, but the "standard" ones need special attention which ultimately make them less general).
1453:{\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)}
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when applying this to fractional order orthogonal integrations/differentiations, the region of integration/differentiation is necessarily of
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2602:<0). In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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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
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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.
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i.e. if going one direction requires a region of integration, then going the other direction requires something like the reverse.
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Hmmm... I think it's fairly obvious what that integration constant is. Even if it wasn't obvious, we would simply use the
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Well, that was pretty straightforward, and it worked. Now, what happens when we exchange the order of composition?
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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and so the inverse operation, differentiation, necessarily has the reverse relations, if composition is to hold.
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I think that the definition via
Laplace transform is incorrect as well. The correct rule for differentiation is
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In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire
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In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire
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3147:{\displaystyle ={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{a}^{t}(t-\tau )^{n-q-1}f(\tau )d\tau }
2905:{\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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Weyl: I don't dispute the fact that it should be given a proper write-up. Whatever gave you this impression?
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2142:{\displaystyle {\mathcal {L}}\left\{{\frac {df}{dt}}\right\}=s\cdot {\mathcal {L}}\left\{f(t)\right\}-f(0)}
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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I don't think Google Books is even a good comparison. The main page for this russian book search is at
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2382:? I think that including a link to a copyrighted work hoping that it's not a problem is a bad idea. β
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A suggestion: Next time give the reason why the link is disputed so people don't have to track it down.
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2258:{\displaystyle {\mathcal {L}}\left\{{\frac {df}{dt}}\right\}=s\cdot {\mathcal {L}}\left\{f(t)\right\}}
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on
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The Weyl version, as I have said before, is something referred to in a major treatise, Zygmund's
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before doing mass systematic removals. This message is updated dynamically through the template
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can someone put the chain rule of differintegration in the basic formal properties section?
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as differentiation is the opposite of integration, the reverse should also hold. that is:
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The integration constant here is clear. Even if it wasn't obvious, we would simply use the
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is needed, not just a constant or set of constants. We call this complementary function "
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3218:, "External links modified" talk page sections are no longer generated or monitored by
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2683:". I'm speaking kind of roughly here, but my point is that you have to consider that:
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If you found an error with any archives or the URLs themselves, you can fix them with
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If we neglected those initialization terms, the last equation would fail our test.
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Please tell me if the NASA paper you cite has been published in a refereed journal.
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https://web.archive.org/web/20040502170831/http://unr.edu/homepage/mcubed/FRG.html
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should be pointed out. If the differintegral is "uninitialized", then although:
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1536:. It should therefore be given a proper write-up. The whole point, according to
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I added a bit of explanation to this extent in the grunwald-leitnikov section.
3224:. No special action is required regarding these talk page notices, other than
826:, a certain oddity about the differintegral should be pointed out. Although:
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dimensional region, it will be an m-th order integration that results in an
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applies equally well to the Weyl differintegral as any other differintegral.
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The process did work successfully. On exchanging the order of composition:
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spatial relations and composition of differintegration w/orthogonal spaces
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Does the composition rule hold always without exception? Consider this:
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http://djvu.504.com1.ru:8019/WWW/c9a77d725039a74f68f7631d14c82f7d.djvu
1464:(Working with a properly initialized differintegral is the subject of
1487:(periodic boundary conditions) is really 'the same as' the others.
2621:, rather than reverting good-faith edits to the article. Thanks,
944:{\displaystyle \mathbb {D} ^{-q}\mathbb {D} ^{q}\neq \mathbb {I} }
291:{\displaystyle \mathbb {D} ^{-q}\mathbb {D} ^{q}\neq \mathbb {I} }
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and furthermore there is the issue i discussed (again, roughly)
1969:{\displaystyle \mathbb {D} ^{\frac {5}{6}}\mathbb {D} ^{2}(t)=0}
2323:. I have removed the link pending clarification of licensing,
874:{\displaystyle \mathbb {D} ^{q}\mathbb {D} ^{-q}=\mathbb {I} }
221:{\displaystyle \mathbb {D} ^{q}\mathbb {D} ^{-q}=\mathbb {I} }
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A question has been raised concerning our ability to link to
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for additional information. I made the following changes:
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tag on the article and an appropriately neutral thread at
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get sued and subsequently pay millions in a settlement
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we consider it as some kind of fractional calculus?
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3214:This message was posted before February 2018.
2015:Error in the definition via Fourier Transform?
813:NB this duplicates the above, but not exactly
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2651:you'll notice that if you integrate an
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1631:Knowledge talk:WikiProject Mathematics
2742:I just wanted to check to see if the
2501:{\displaystyle \mathbb {D} _{t}^{q}.}
1901:{\displaystyle \mathbb {D} ^{2}(t)=0}
820:initialization of the differintegrals
809:initialization of the differintegrals
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2421:file is legitimately licensed.
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1546:fundamental formal properties
109:and see a list of open tasks.
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1559:page give reference to this.
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803:oddity.. almost duplicate..
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169:A certain oddity about the
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3245:(last update: 5 June 2024)
3167:Hello fellow Wikipedians,
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1861:However, it is clear that
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1606:07:13, 3 August 2010 (UTC)
2939:is the fractional order?
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2403:http://www.poiskknig.ru/
2315:Copyright status of link
2292:Merger? and applications
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1625:ImportantLabeledEquation
1580:17:04, 23 Apr 2004 (UTC)
1520:23:39, 22 Apr 2004 (UTC)
141:project's priority scale
3163:External links modified
2543:operator. Applied to a
98:WikiProject Mathematics
3148:
3004:
2933:
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2663:dimensional function.
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2259:
2143:
1970:
1902:
1852:
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1248:complementary function
1214:
1097:
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592:complimentary function
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444:
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28:This article is rated
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3005:
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1542:mathematical analysis
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1263:{\displaystyle \Psi }
1215:
1098:
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876:
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562:
445:
293:
223:
3226:regular verification
3015:
2953:
2923:
2773:
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2698:fractional dimension
2562:
2475:
2378:Plus, didn't Google
2300:those in the know.
2179:
2048:
1919:
1868:
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1534:Trigonometric Series
1277:
1254:
1225:initialization terms
1113:
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903:
833:
601:
572:initialization terms
460:
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121:mathematics articles
3216:After February 2018
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2551:-differintegral of
2529:applied mathematics
2525:fractional calculus
2513:fractional calculus
2494:
2466:fractional calculus
2000:I tried explaining
1544:, is to get beyond
1485:Weyl differintegral
1481:fractional calculus
1382:
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824:fractional calculus
715:
629:
3270:InternetArchiveBot
3221:InternetArchiveBot
3144:
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2139:
1966:
1898:
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1210:
1093:
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818:Before discussing
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288:
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90:Mathematics portal
34:content assessment
3246:
3158:
3157:
3075:
3046:
2932:{\displaystyle q}
2915:
2914:
2833:
2804:
2755:{\displaystyle n}
2464:operator used in
2456:is the combined
2437:Which is clearer?
2273:orthogonal system
2211:
2171:differintegrating
2080:
2038:
2026:comment added by
1936:
1846:
1842:
1825:
1787:
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1609:
1592:comment added by
1576:not heriarchial.
1366:
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1241:dynamical systems
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2277:orthogonal basis
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1718:Composition rule
1608:
1586:
1552:Charles Matthews
1510:unsubstantiated.
1491:Charles Matthews
1473:Charles Matthews
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797:Charles Matthews
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3230:have permission
3220:
3187:
3180:this simple FaQ
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2681:with respect to
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2537:differentiation
2535:is a combined
2473:
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2458:differentiation
2439:
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2153:195.113.191.162
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1939:
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1635:Oleg Alexandrov
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2350:Moonriddengirl
2336:Moonriddengirl
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2302:208.120.110.46
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2014:
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1525:NASA citation
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232:
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137:Low-priority
136:
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62:Lowβpriority
40:WikiProjects
2722:chain rule?
2541:integration
2462:integration
2450:mathematics
2022:βPreceding
1648:Fresheneesz
1588:βPreceding
1540:, of doing
1538:G. H. Hardy
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
3290:Categories
3277:Report bug
2730:Kevin Baas
2714:Kevin Baas
2638:Kevin Baas
2623:Le Docteur
2519:Option 2:
2444:Option 1:
2332:WP:LINKVIO
2281:Kevin Baas
2006:Kevin Baas
1991:Kevin Baas
1912:Therefore
1707:Kevin Baas
1698:Kevin Baas
1662:Kevin Baas
1616:Kevin Baas
1594:Wadewizard
1578:Kevin Baas
1518:Kevin Baas
231:(That is,
3260:this tool
3253:this tool
3190:dead link
2269:transform
884:That is,
3266:Cheers.β
2545:function
2366:DreamGuy
2024:unsigned
1602:contribs
1590:unsigned
1227:such as
574:such as
3194:tag to
3176:my edit
2547:Ζ, the
890:inverse
237:inverse
139:on the
3186:Added
2619:WT:WPM
2612:expert
2531:, the
2452:, the
1682:Joriki
158:Oddity
36:scale.
2597:: -->
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2328:WT:EL
2762:in
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3292::
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