Fractional integration and differentiation

An extension of the operations of integration and differentiation to the case of fractional powers. Let be integrable on the interval , let be the integral of along , while is the integral of along , . One then has

(1)

where is the gamma-function. The right-hand side makes sense for every . The relation (1) defines the fractional integral (or the Riemann–Liouville integral) of order of with starting point . The operator was studied by B. Riemann (1847) for complex values of the parameter . The operator is linear and has the semi-group property:

The operation inverse to fractional integration is known as fractional differentiation: If , then is the fractional derivative of order of . If , Marchaut's formula applies:

The concept of fractional integration and differentiation was first introduced by J. Liouville (1832); he studied, in particular, the operator , :

(subject to appropriate restrictions on ; cf. [1], which also contains estimates of the operator in ).

The following definition (H. Weyl, 1917) is convenient for an integrable -periodic function with zero average value over the period. If

then the Weyl integral of order of is defined by the formula

(2)

and the derivative of order is defined by the equation

where is the smallest integer larger than (it should be noted that coincides with ).

These definitions were further developed in the framework of the theory of generalized functions. For periodic generalized functions

the operation of fractional integration is realized according to formula (2) for all real (if is negative, coincides with the partial derivative of order ) and has the semi-group property with respect to the parameter .

In an -dimensional space the analogue of the operator of fractional integration is the Riesz potential (or the integral of potential type)

The operation inverse to is said to be the Riesz derivative of order .

References

[1]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[2]	A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
[3]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[4]	M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)