Fractional Differentiation

Abstract

From mathematical point of view, fractional derivativea f (ν) (x) of order ν is a function of three variables: the lower limit a, the argument x and the order ν. Naming this functional the derivative, we believe that in case of integer ν, ν = n, it coincides with the n-order derivative. Extending the interrelation d f (n) (x)/dx = f (n+1) (x) to negative values of order, we interpret f (−m) (x), m > 0, as integrals, and f (0) (x) = f(x). Now, the functiona f (ν) (x) can be considered as an analytic continuation of f (n) (x), n = …, − 2, − 1,0, 1, 2,… saving basic properties of multiple derivatives.