In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator.
Applied to a function ƒ, the q-differintegral of f, here denoted by is the fractional derivative (if q > 0) or 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.
The four most common forms are: The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide.
[1] They can be represented via Laplace, Fourier transforms or via Newton series expansion.
Recall the continuous Fourier transform, here denoted
( ω ) =
2 π
− i ω t
{\displaystyle F(\omega )={\mathcal {F}}\{f(t)\}={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,dt}
Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
= i ω
{\displaystyle {\mathcal {F}}\left[{\frac {df(t)}{dt}}\right]=i\omega {\mathcal {F}}[f(t)]}
( i ω
Under the bilateral Laplace transform, here denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.")
from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{L}} and defined as
{\textstyle {\mathcal {L}}[f(t)]=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt}
, differentiation transforms into a multiplication
{\displaystyle {\mathcal {L}}\left[{\frac {df(t)}{dt}}\right]=s{\mathcal {L}}[f(t)].}
Generalizing to arbitrary order and solving for
Representation via Newton series is the Newton interpolation over consecutive integer orders:
For fractional derivative definitions described in this section, the following identities hold:
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;[3] this forms part of the decision making process on which one to choose: