Differintegral


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 or fractional integral. 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:
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 bilateral Laplace transform, here denoted by and defined as, differentiation transforms into a multiplication
Generalizing to arbitrary order and solving for Dqf, one obtains

Basic formal properties

Linearity rules
Zero rule
Product rule
In general, composition rule is not satisfied:

A selection of basic formulae