See the page on fractional calculus for the general context.

This class of functions are important in the theory of fractional calculus.

This is achieved by analog or digital implementations of fractional calculus.

It covers research on fractional calculus, special functions, integral transforms, and some closely related areas of applied analysis.

By means of the Fourier transform, pseudo-differential operators can be defined which allow for fractional calculus.

In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus.

In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator.

Working with a properly initialized differintegral is the subject of initialized fractional calculus.

Before discussing 'initialization of the differintegrals' in fractional calculus, a certain oddity about the differintegral should be pointed out.

Fractional calculus - a branch of analysis that studies the possibility of taking real or complex powers of the differentiation operator.

Fractional Calculus and Applied Analysis, vol.

Fractional calculus on time scales is treated in Bastos, Mozyrska, and Torres.

In fractional calculus, this formula can be used to construct a notion of differintegral, allowing one to differentiate or integrate a fractional number of times.

Related applications include Acoustical wave equations (see also the Applications section in the Fractional Calculus article).

In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complementary function is needed, not just a constant or set of constants.

Heaviside went farther, and defined fractional power of , thus establishing a connection between operational calculus and fractional calculus.

Fractional dynamics - investigates the behaviour of objects and systems that are described by differentiation and integration of fractional orders using methods of fractional calculus.

The usage of fractional elements for description of ideal Bode's control loop is one of the most promising applications of fractional calculus in the process control field.

In addition to n-th derivatives for any natural number n, there are various ways to define derivatives of fractional or negative orders, which are studied in fractional calculus.

In mathematics, the Grünwald-Letnikov derivative is a basic extension of the derivative in fractional calculus, that allows one to take the derivative a non-integer number of times.

In mathematics, the Weyl integral is an operator defined, as an example of fractional calculus, on functions f on the unit circle having integral 0 and a Fourier series.

"Multiresolution analysis with sampling subspaces," (jointly with G. Walter), Journal of Fractional Calculus and Applied Analysis, Vol.

The Riemann-Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832.

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models Imperial College Press, 2010.

"Fractional Wigner distribution and ambiguity functions," (jointly with V. B. Shakhmurov) Journal of Fractional Calculus and Applied Analysis, Vol.