Two Definitions of Fractional Derivative of Powers Functions

Abstract

We consider the set of powers functions defined on and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.

Definitions of Real Order Integrals and Derivatives Using Operator Approach

“…As a final conclusion, we would like to say that the approach by means of integral operators and derivative operators derived from integral operators is better and more general than the two first ones that we have given in our work [5] because the properties of the set š of functions › we are looking for (for instance causal function of the real variable †) are assumed to be known from the beginning. The limit of the applicability of the method is then clear.…”

“…Since then, several approaches have been done [1], [2], [3], [4] In reference [5], we have considered the case of -and for . / $ .…”