Fractional calculus

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In mathematics, fractional calculus is a branch of mathematical analysis, studying the possibility of taking real number powers of the differential operator

D = d/dx

and the integration operator I. For example, one may pose the question of interpreting meaningfully

√D = D^½

as a square root of the differentiation operator, qua operator. That means, some operator that when applied twice to a function, will have the same effect as differentiation. More generally, one can look at the question of defining

D^s

for real number values of s, in such a way that when s takes an integer value n, the usual power of n-fold differentiation is recovered for n > 0, and the −nth power of I when n < 0.

There are various possible reasons for looking at this question. One is that in this way the semigroup of powers Dⁿ in the discrete variable n is seen inside a continuous semigroup (one hopes) with parameter s which is a real number. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that fraction is then a misnomer for the exponent, but the fractional calculus name has become traditional.

As far as the existence of such a theory is concened, the foundations of the subject were laid by Liouville in a paper from 1832. The fractional derivative of a function to order a is often now defined by means of the Fourier transform. An important point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integral cases we can't say that the fractional derivative at x of a function f depends only on the graph of f very near x, in gthe way that integer-power derivatives certainly do. Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out. To use a metaphor, the fractional derivative requires some peripheral vision.

The classical form of fractional calculus is given by the Riemann-Liouville differintegral. The theory for periodic functions, therefore including the 'boundary condition' of repeating after a period, is the Weyl differintegral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (so, applies to functions on the unit circle integrating to 0).

In the context of functional analysis, functions f(D) more general than powers are studied in the functional calculus of spectral theory. The theory of pseudo-differential operators also allows one to consider powers of D. The operators arising are examples of singular integral operators; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potientials. So there are a number of contemporary theories available, within which fractional calculus can be discussed.