Definitions
Nearby Words

# Differintegral

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

$mathbb\left\{D\right\}^q_t.$

See the page on fractional calculus for the general context.

## Standard definitions

The three most common forms are:

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.
$\left\{\right\}_amathbb\left\{D\right\}^q_tf\left(t\right)$ $=frac\left\{d^qf\left(t\right)\right\}\left\{d\left(t-a\right)^q\right\}$
$=frac\left\{1\right\}\left\{Gamma\left(n-q\right)\right\} frac\left\{d^n\right\}\left\{dt^n\right\} int_\left\{a\right\}^\left\{t\right\}\left(t-tau\right)^\left\{n-q-1\right\}f\left(tau\right)dtau$

$\left\{\right\}_amathbb\left\{D\right\}^q_tf\left(t\right)$ $=frac\left\{d^qf\left(t\right)\right\}\left\{d\left(t-a\right)^q\right\}$
$=lim_\left\{N to infty\right\}left\left[frac\left\{t-a\right\}\left\{N\right\}right\right]^\left\{-q\right\}sum_\left\{j=0\right\}^\left\{N-1\right\}\left(-1\right)^j\left\{q choose j\right\}fleft\left(t-jleft\left[frac\left\{t-a\right\}\left\{N\right\}right\right]right\right)$

This is formally similar to the Riemann-Liouville differintegral, but applies to periodic functions, with integral zero over a period.

## Definitions via transforms

Recall the continuous Fourier transform, here denoted $mathcal\left\{F\right\}$ :

$F\left(omega\right) = mathcal\left\{F\right\}\left\{f\left(t\right)\right\} = frac\left\{1\right\}\left\{sqrt\left\{2pi\right\}\right\}int_\left\{-infty\right\}^infty f\left(t\right) e^\left\{- iomega t\right\},dt$

Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:

$mathcal\left\{F\right\}left\left[mathbb\left\{D\right\}f\left(t\right)right\right] = mathcal\left\{F\right\}left\left[frac\left\{df\left(t\right)\right\}\left\{dt\right\}right\right] = i omega mathcal\left\{F\right\}\left[f\left(t\right)\right]$

So,

$mathbb\left\{D\right\}f\left(t\right) = mathcal\left\{F\right\}^\left\{-1\right\}left\left\{\left(i omega\right)mathcal\left\{F\right\}\left[f\left(t\right)\right]right\right\}$

which generalizes to

$mathbb\left\{D\right\}^qf\left(t\right)=mathcal\left\{F\right\}^\left\{-1\right\}left\left\{\left(i omega\right)^qmathcal\left\{F\right\}\left[f\left(t\right)\right]right\right\}.$

Under the Laplace transform, here denoted by $mathcal\left\{L\right\}$, differentiation transforms into a multiplication

$mathcal\left\{L\right\}left\left[frac\left\{df\left(t\right)\right\}\left\{dt\right\}right\right] = smathcal\left\{L\right\}\left[f\left(t\right)\right].$

Generalizing to arbitrary order and solving for Dqf(t), one obtains

$mathbb\left\{D\right\}^qf\left(t\right)=mathcal\left\{L\right\}^\left\{-1\right\}left\left\{s^\left\{q\right\}mathcal\left\{L\right\}\left[f\left(t\right)\right]right\right\}.$

## Basic formal properties

Linearity rules

$mathbb\left\{D\right\}^\left\{q\right\}\left(x+y\right)=mathbb\left\{D\right\}^\left\{q\right\}\left(x\right)+mathbb\left\{D\right\}^\left\{q\right\}\left(y\right)$
$mathbb\left\{D\right\}^\left\{q\right\}\left(ax\right)=amathbb\left\{D\right\}^\left\{q\right\}\left(x\right)$

Composition (or semigroup) rule

$mathbb\left\{D\right\}^amathbb\left\{D\right\}^\left\{b\right\}x = mathbb\left\{D\right\}^\left\{a+b\right\}x$

Zero rule

$mathbb\left\{D\right\}^\left\{0\right\}x=x$

Subclass rule

$mathbb\left\{D\right\}^\left\{a\right\}x=d^\left\{a\right\}x$ for a a natural number

Product rule of differintegration

$mathbb\left\{D\right\}^q_t\left(xy\right)=sum_\left\{j=0\right\}^\left\{infty\right\} \left\{q choose j\right\}mathbb\left\{D\right\}^j_t\left(x\right)mathbb\left\{D\right\}^\left\{q-j\right\}_t\left(y\right)$

## Some basic formulae

$mathbb\left\{D\right\}^\left\{q\right\}\left(t^n\right)=frac\left\{Gamma\left(n+1\right)\right\}\left\{Gamma\left(n+1-q\right)\right\}t^\left\{n-q\right\}$
$mathbb\left\{D\right\}^\left\{q\right\}\left(sin\left(t\right)\right)=sin left\left(t+frac\left\{qpi\right\}\left\{2\right\} right\right)$
$mathbb\left\{D\right\}^\left\{q\right\}\left(e^\left\{at\right\}\right)=a^\left\{q\right\}e^\left\{at\right\}$

## References

• "An Introduction to the Fractional Calculus and Fractional Differential Equations", by Kenneth S. Miller, Bertram Ross (Editor), John Wiley & Sons; 1 edition (May 19, 1993). ISBN 0-471-58884-9.
• "The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V)", by Keith B. Oldham, Jerome Spanier, Academic Press; (November 1974). ISBN 0-12-525550-0.
• "Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications", (Mathematics in Science and Engineering, vol. 198), by Igor Podlubny, Academic Press (October 1998). ISBN 0-12-558840-2.
• "Fractals and Fractional Calculus in Continuum Mechanics", by A. Carpinteri (Editor), F. Mainardi (Editor), Springer-Verlag Telos; (January 1998). ISBN 3-211-82913-X.
• "Physics of Fractal Operators", by Bruce J. West, Mauro Bologna, Paolo Grigolini, Springer Verlag; (January 14, 2003). ISBN 0-387-95554-2
• Operator of fractional derivative in the complex plane, by Petr Zavada, Commun.Math.Phys.192, pp. 261-285,1998. doi:10.1007/s002200050299 (available online or as the arXiv preprint)
• Relativistic wave equations with fractional derivatives and pseudodifferential operators, by Petr Zavada, Journal of Applied Mathematics, vol. 2, no. 4, pp. 163-197, 2002. doi:10.1155/S1110757X02110102 (available online or as the arXiv preprint)