Editing Fractional calculus (section)

{{Short description|Branch of mathematical analysis}}
{{For|the associated operator|differintegral}}
{{Calculus}}
'''Fractional calculus''' is a branch of [[mathematical analysis]] that studies the several different possibilities of defining [[real number]] powers or [[complex number]] powers of the [[derivative|differentiation]] [[operator (mathematics)|operator]] <math>D</math> 
<math display="block">D f(x) = \frac{d}{dx} f(x)\,,</math>

and of the [[integral|integration]] operator <math>J</math> <ref group=Note>The symbol <math>J</math> is commonly used instead of the intuitive <math>I</math> in order to avoid confusion with other concepts identified by similar {{nowrap|<math>I</math>–like}} [[glyph]]s, such as [[identity (mathematics)|identities]].</ref>
<math display="block">J f(x) = \int_0^x f(s) \,ds\,,</math>

and developing a [[calculus]] for such operators generalizing the classical one.

In this context, the term ''powers'' refers to iterative application of a [[linear operator]] <math>D</math> to a [[function (mathematics)|function]] {{nowrap|<math>f</math>,}} that is, repeatedly [[function composition|composing]] <math>D</math> with itself, as in
<math display="block">\begin{align}
D^n(f) &= (\underbrace{D\circ D\circ D\circ\cdots \circ D}_n)(f) \\
       &= \underbrace{D(D(D(\cdots D}_n (f)\cdots))).
\end{align}</math>

For example, one may ask for a meaningful interpretation of
<math display="block">\sqrt{D} = D^{\scriptstyle{\frac12}}</math>

as an analogue of the [[functional square root]] for the differentiation operator, that is, an expression for some linear operator that, when applied {{em|twice}} to any function, will have the same effect as [[derivative|differentiation]]. More generally, one can look at the question of defining a linear operator
<math display="block">D^a</math>

for every real number <math>a</math> in such a way that, when <math>a</math> takes an [[integer]] value {{nowrap|<math>n\in\mathbb{Z}</math>,}} it coincides with the usual {{nowrap|<math>n</math>-fold}} differentiation <math>D</math> if {{nowrap|<math>n>0</math>,}} and with the {{nowrap|<math>n</math>-th}} power of <math>J</math> when {{nowrap|<math>n<0</math>.}}

One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator <math>D</math> is that the [[set (mathematics)|sets]] of operator powers <math>\{D^a\mid a\in\R\}</math> defined in this way are ''continuous'' [[semigroup]]s with parameter {{nowrap|<math>a</math>,}} of which the original ''discrete'' semigroup of <math>\{D^n\mid n\in\Z\}</math> for integer <math>n</math> is a [[denumerable set|denumerable]] subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.

Fractional [[differential equation]]s, also known as extraordinary differential equations,<ref name=Zwillinger2014>{{cite book |author=Daniel Zwillinger |title=Handbook of Differential Equations |url=https://books.google.com/books?id=9QLjBQAAQBAJ |date=12 May 2014 |publisher=Elsevier Science |isbn=978-1-4832-2096-3}}</ref> are a generalization of differential equations through the application of fractional calculus.