Differintegral
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Calculus  



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In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the qdifferintegral of f, here denoted by
is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the qth 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 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.
 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.
Definitions via transforms
Recall the continuous Fourier transform, here denoted :
Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
So,
which generalizes to
Under the Laplace transform, here denoted by , differentiation transforms into a multiplication
Generalizing to arbitrary order and solving for D^{q}f(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 formulæ
See also
References
 ↑ See Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier. pp. 75 (Property 2.4).
 "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 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, 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, Academic Press (October 1998). ISBN 0125588402.
 "Fractals and Fractional Calculus in Continuum Mechanics", by A. Carpinteri (Editor), F. Mainardi (Editor), SpringerVerlag Telos; (January 1998). ISBN 321182913X.
 Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. by F. Mainardi, Imperial College Press, 2010. 368 pages.
 Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. by V.E. Tarasov, Springer, 2010. 450 pages.
 Fractional Derivatives for Physicists and Engineers by V.V. Uchaikin, Springer, Higher Education Press, 2012, 385 pages.
 "Physics of Fractal Operators", by Bruce J. West, Mauro Bologna, Paolo Grigolini, Springer Verlag; (January 14, 2003). ISBN 0387955542
External links
 MathWorld – Fractional calculus
 MathWorld – Fractional derivative
 Specialized journal: Fractional Calculus and Applied Analysis (19982014) and Fractional Calculus and Applied Analysis (from 2015)
 Specialized journal: Fractional Differential Equations (FDE)
 Specialized journal: Progress in Fractional Differentiation and Applications
 Specialized journal: Communications in Fractional Calculus (ISSN 22183892)
 Specialized journal: Journal of Fractional Calculus and Applications (JFCA)
 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.
 Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, 367–386. (available as original article, or preprint at Arxiv.org)
 Operator of fractional derivative in the complex plane, by P. Zavada in Commun.Math.Phys. 192 (1998) 261285, or available as the arXiv ePrint