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In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator. Applied to a function ?, 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 legitimate definitions of the differintegral.

Standard definitions

The four 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. Here, .

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

In opposite to the Riemann-Liouville differintegral, Caputo derivative of a constant 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 .

Definitions via transforms

Recall the continuous Fourier transform, here denoted  :

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


which generalizes to

Under the bilateral Laplace transform, here denoted by and defined as , differentiation transforms into a multiplication

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

Basic formal properties

Linearity rules

Zero rule

Product rule

In general, composition (or semigroup) rule is not satisfied[1]:

A selection of basic formulae

See also


  1. ^ See Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). "2. Fractional Integrals and Fractional Derivatives §2.1 Property 2.4". Theory and Applications of Fractional Differential Equations. Elsevier. p. 75. ISBN 9780444518323.

External links

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