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{{redirect-distinguish|Fractional integration|Autoregressive fractionally integrated moving average}}
{{Calculus|expanded=Specialized calculi}}

In [[fractional calculus]], an area of [[mathematical analysis]], the '''differintegral''' is a combined  [[Differential operator|differentiation]]/[[integral operator|integration]] operator. Applied to a [[function (mathematics)|function]] ƒ, the ''q''-differintegral of ''f'', here denoted by
:<math>\mathbb{D}^q f</math>
is the [[Fractional_calculus#Historical_notes|fractional derivative]] (if ''q'' > 0) or [[Fractional_calculus#Fractional_integrals|fractional integral]] (if ''q'' < 0).  If ''q'' = 0, then the ''q''-th differintegral of a function is the function itself.  In the context of fractional integration and differentiation, there are several definitions of the differintegral.

==Standard definitions==

The four most common forms are:

*The [[Riemann–Liouville differintegral]]{{pb}}This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the [[Cauchy formula for repeated integration]] to arbitrary order. Here, <math>n = \lceil q \rceil</math>. <math display="block"> \begin{align}
{}^{RL}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\
& =\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{a}^t (t-\tau)^{n-q-1}f(\tau)d\tau
\end{align}</math>
*The [[Grunwald–Letnikov differintegral]]{{pb}}The Grunwald–Letnikov differintegral is a direct generalization of the definition of a [[derivative]].  It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot. <math display="block">\begin{align}
{}^{GL}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\
& =\lim_{N \to \infty}\left[\frac{t-a}{N}\right]^{-q}\sum_{j=0}^{N-1}(-1)^j{q \choose j}f\left(t-j\left[\frac{t-a}{N}\right]\right)
\end{align}</math>
*The [[Weyl differintegral]]{{pb}} This is formally similar to the Riemann–Liouville differintegral, but applies to [[periodic function]]s, with integral zero over a period.
*The [[Caputo differintegral]]{{pb}}In opposite to the Riemann-Liouville differintegral, Caputo derivative of a constant <math>f(t)</math> is equal to zero. Moreover, a form of the Laplace transform allows to simply evaluate the initial conditions by computing finite, integer-order derivatives at point <math>a</math>. <math display="block">\begin{align}
{}^{C}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\
& =\frac{1}{\Gamma(n-q)} \int_{a}^t \frac{f^{(n)}(\tau)}{(t-\tau)^{q-n+1}}d\tau
\end{align}</math>

==Definitions via transforms==

The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide.<ref>{{cite book |url=https://books.google.com/books?id=mPXzp1f7ycMC&pg=PA11 |first=Richard |last=Herrmann|title=Fractional Calculus: An Introduction for Physicists |year=2011 |isbn=9789814551076 }}</ref> They can be represented via Laplace, Fourier transforms or via Newton series expansion.

Recall the [[continuous Fourier transform]], here denoted <math> \mathcal{F}</math>:
<math display="block"> F(\omega) =  \mathcal{F}\{f(t)\} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) e^{- i\omega t}\,dt </math>

Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
<math display="block">\mathcal{F}\left[\frac{df(t)}{dt}\right] = i \omega \mathcal{F}[f(t)]</math>

So,
<math display="block">\frac{d^nf(t)}{dt^n} = \mathcal{F}^{-1}\left\{(i \omega)^n\mathcal{F}[f(t)]\right\}</math>
which generalizes to
<math display="block">\mathbb{D}^qf(t) = \mathcal{F}^{-1}\left\{(i \omega)^q\mathcal{F}[f(t)]\right\}.</math>

Under the [[bilateral Laplace transform]], here denoted by <math> \mathcal{L}</math> and defined as <math display="inline"> \mathcal{L}[f(t)] =\int_{-\infty}^\infty e^{-st} f(t)\, dt</math>, differentiation transforms into a multiplication
<math display="block">\mathcal{L}\left[\frac{df(t)}{dt}\right] = s\mathcal{L}[f(t)].</math>

Generalizing to arbitrary order and solving for <math> \mathbb{D}^qf(t)</math>, one obtains
<math display="block">\mathbb{D}^qf(t)=\mathcal{L}^{-1}\left\{s^q\mathcal{L}[f(t)]\right\}.</math>

Representation via Newton series is the Newton interpolation over consecutive integer orders:

<math display="block">\mathbb{D}^qf(t) =\sum_{m=0}^{\infty} \binom {q}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x).</math>

For fractional derivative definitions described in this section, the following identities hold:

:<math>\mathbb{D}^q(t^n)=\frac{\Gamma(n+1)}{\Gamma(n+1-q)}t^{n-q}</math>
	
:<math>\mathbb{D}^q(\sin(t))=\sin \left( t+\frac{q\pi}{2} \right) </math>
	
:<math>\mathbb{D}^q(e^{at})=a^q e^{at}</math><ref>See {{cite book |page=16 |url=https://books.google.com/books?id=mPXzp1f7ycMC&pg=PA11 |first=Richard |last=Herrmann|title=Fractional Calculus: An Introduction for Physicists | year=2011 |isbn=9789814551076 }}</ref>

==Basic formal properties==

*''[[Linear operator|Linearity]] rules'' <math display="block">\mathbb{D}^q(f+g) = \mathbb{D}^q(f)+\mathbb{D}^q(g)</math>
<math display="block">\mathbb{D}^q(af) = a\mathbb{D}^q(f)</math>
*''Zero rule'' <math display="block">\mathbb{D}^0 f = f </math>
*''Product rule'' <math display="block">\mathbb{D}^q_t(fg) = \sum_{j=0}^{\infty} {q \choose j}\mathbb{D}^j_t(f)\mathbb{D}^{q-j}_t(g)</math>

In general, ''composition (or [[semigroup]]) rule'' is a desirable property, but is hard to achieve mathematically and hence is '''not always completely satisfied''' by each proposed operator;<ref>See {{cite book |page=75 |chapter=2. Fractional Integrals and Fractional Derivatives §2.1 Property 2.4 |chapter-url=https://books.google.com/books?id=uxANOU0H8IUC&pg=PA75 |first1=A. A. |last1=Kilbas |first2=H. M. |last2=Srivastava |first3=J. J. |last3=Trujillo |title=Theory and Applications of Fractional Differential Equations |publisher=Elsevier |year=2006 |isbn=9780444518323 }}</ref> this forms part of the decision making process on which one to choose:

* <math display="inline">\mathbb{D}^a\mathbb{D}^{b}f = \mathbb{D}^{a+b}f</math> (ideally)
* <math display="inline">\mathbb{D}^a\mathbb{D}^{b}f \neq \mathbb{D}^{a+b}f</math> (in practice)

==See also==
* [[Fractional-order integrator]]

==References==
{{Reflist}}
{{refbegin}}
*{{cite book |first=Kenneth S. |last=Miller |editor-first=Bertram |editor-last=Ross |title=An Introduction to the Fractional Calculus and Fractional Differential Equations |publisher=Wiley |year=1993 |isbn=0-471-58884-9 }}
*{{cite book |first1=Keith B. |last1=Oldham |first2=Jerome |last2=Spanier |title=The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order |publisher=Academic Press |series=Mathematics in Science and Engineering |volume=V |year=1974 |isbn=0-12-525550-0 }}
*{{cite book |first=Igor |last=Podlubny |title=Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications |publisher=Academic Press  |series=Mathematics in Science and Engineering |volume=198 |year=1998 |isbn=0-12-558840-2 }}
*{{cite book |editor-first=A. |editor-last=Carpinteri |editor2-first=F. |editor2-last=Mainardi |title=Fractals and Fractional Calculus in Continuum Mechanics |publisher=Springer-Verlag |year=1998 |isbn=3-211-82913-X }}
*{{cite book |first=F. |last=Mainardi |title=Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models |publisher=Imperial College Press |year=2010 |isbn=978-1-84816-329-4 |url=http://www.worldscibooks.com/mathematics/p614.html |archive-url=https://web.archive.org/web/20120519174508/http://www.worldscibooks.com/mathematics/p614.html |url-status=dead |archive-date=2012-05-19 }}
*{{cite book |first=V.E. |last=Tarasov |title=Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media |publisher=Springer |year=2010 |isbn=978-3-642-14003-7 |url=https://www.springer.com/physics/complexity/book/978-3-642-14003-7|series=Nonlinear Physical Science }}
*{{cite book |first=V.V. |last=Uchaikin |title=Fractional Derivatives for Physicists and Engineers |publisher=Springer |year=2012 |isbn=978-3-642-33910-3 |url=https://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-642-33910-3|series=Nonlinear Physical Science |bibcode=2013fdpe.book.....U }}
*{{cite book |first1=Bruce J. |last1=West |first2=Mauro |last2=Bologna |first3=Paolo |last3=Grigolini |title=Physics of Fractal Operators |publisher=Springer Verlag |year=2003 |isbn=0-387-95554-2 |url=https://books.google.com/books?id=EgyTpQZOga0C&pg=PR7}}
{{refend}}

==External links==
* [http://mathworld.wolfram.com/FractionalCalculus.html MathWorld – Fractional calculus]
*[http://mathworld.wolfram.com/FractionalDerivative.html MathWorld – Fractional derivative]
*Specialized journal: [http://www.diogenes.bg/fcaa/ Fractional Calculus and Applied Analysis (1998-2014)] and [http://www.degruyter.com/view/j/fca Fractional Calculus and Applied Analysis (from 2015)]
*Specialized journal: [https://archive.today/20120712033445/http://fde.ele-math.com/ Fractional Differential Equations (FDE)]
*Specialized journal: [https://web.archive.org/web/20180421124535/http://www.nonlinearscience.com/journal_2218-3892.php Communications in Fractional Calculus] ({{issn|2218-3892}})
* Specialized journal: [http://fcag-egypt.com/Journals/JFCA/ Journal of Fractional Calculus and Applications (JFCA)]
*{{cite web |first1=Carl F. |last1=Lorenzo |first2=Tom T. |last2=Hartley |title=Initialized Fractional Calculus |date=2002 |work=Information Technology |publisher=Tech Briefs Media Group |url=https://www.techbriefs.com/component/content/article/tb/techbriefs/information-sciences/2264}}
* https://web.archive.org/web/20040502170831/http://unr.edu/homepage/mcubed/FRG.html
* [http://www.tuke.sk/podlubny/fc_resources.html Igor Podlubny's collection of related books, articles, links, software, etc. ]
*{{cite journal |first=I. |last=Podlubny |title=Geometric and physical interpretation of fractional integration and fractional differentiation |journal=Fractional Calculus and Applied Analysis |volume=5 |issue=4 |pages=367–386 |year=2002 |url=http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf |arxiv=math.CA/0110241 |bibcode=2001math.....10241P |access-date=2004-05-18 |archive-date=2006-04-07 |archive-url=https://web.archive.org/web/20060407100616/http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf |url-status=dead }}
*{{cite journal |first=P. |last=Zavada |title=Operator of fractional derivative in the complex plane |journal= Communications in Mathematical Physics|volume=192 |issue= 2|pages=261–285 |year=1998 |doi=10.1007/s002200050299 |arxiv=funct-an/9608002|bibcode=1998CMaPh.192..261Z |s2cid=1201395 }}

[[Category:Fractional calculus]]
[[Category:Generalizations of the derivative]]
[[Category:Linear operators in calculus]]