In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus. This operator is here denoted Mathematics is the study of quantity, structure, space and change. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In mathematics, fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator D = d/dx and the integration operator I. In this context powers refer iterative application, in the same sense that f2(x) = f(f(x)). For example...

See the page on fractional calculus for the general context.

Basic formal properties

Linearity rules

Composition or semigroup rule In mathematics, a semigroup is a set with an associative binary operation on it. ...

Zero rule

Subclass rule

for a a natural number

Product rule of differintegration Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

Some basic formulae

Standard definitions

The three most common forms are:

• The Riemann-Liouville differintegral

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.

definition In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly. ...

• The Grunwald-Letnikov differintegral

definition In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly. ...

• The Weyl differintegral

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

In mathematics, a periodic function is a function that repeats its values, after adding some definite period to the variable. ...

Definitions via transforms

Using the continuous Fourier transform, here denoted F: in Fourier space, differentiation transforms into a multiplication: In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

This generalizes to

definition

Under the Laplace transform, here denoted by L, differentiation transforms to a multiplication In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by: The lower limit of 0− is short notation to mean and assures the inclusion of the entire dirac delta function...

Generalizing to arbitrary order and solving for D^qf(t), one obtains

definition

External links

• MathWorld - Fractional calculus (http://mathworld.wolfram.com/FractionalCalculus.html)
• MathWorld - Fractional derivative (http://mathworld.wolfram.com/FractionalDerivative.html)
• Specialized journal: Fractional Calculus and Applied Analysis (http://www.diogenes.bg/fcaa/)
• http://www.nasatech.com/Briefs/Oct02/LEW17139.html
• http://unr.edu/homepage/mcubed/FRG.html
• Igor Podlubny's collection of related books, articles, links, software, etc.  (http://www.tuke.sk/podlubny/fc_resources.html)
• Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis (http://www.diogenes.bg/fcaa/), vol. 5, no. 4, 2002, 367–386. (available as original article (http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf), or preprint at Arxiv.org (http://arxiv.org/abs/math.CA/0110241))

Resource books

"An Introduction to the Fractional Calculus and Fractional Differential Equations"

by Kenneth S. Miller, Bertram Ross (Editor)

Hardcover: 384 pages ; Dimensions (in inches): 1.00 x 9.75 x 6.50

Publisher: John Wiley & Sons; 1 edition (May 19, 1993)

ISBN 0471588849

"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

Hardcover

Publisher: Academic Press; (November 1974)

ISBN 0125255500

"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

Hardcover

Publisher: Academic Press; (October 1998)

ISBN 0125588402

"Fractals and Fractional Calculus in Continuum Mechanics"

by A. Carpinteri (Editor), F. Mainardi (Editor)

Paperback: 348 pages

Publisher: Springer-Verlag Telos; (January 1998)

ISBN 321182913X

"Physics of Fractal Operators"

by Bruce J. West, Mauro Bologna, Paolo Grigolini

Hardcover: 368 pages

Publisher: Springer Verlag; (January 14, 2003)

ISBN 0387955542