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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 legitimate definitions of the differintegral.

Standard definitions[edit]

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[edit]

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[edit]

Linearity rules

Zero rule

Product rule

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

See also[edit]


  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[edit]