Real and complex order integrals and derivatives operators over the set of causal functions

TL;DR

This paper develops a unified operator-based framework for fractional integrals and derivatives over causal functions, extending classical definitions and comparing with existing approaches using properties of special functions.

Contribution

It introduces a unified definition of fractional integrals and derivatives based on the sign of the order's real part, expanding the operator approach for causal functions.

Findings

Properties of fractional integrals are established.

Fractional derivatives are derived from integrals.

Comparison with classical definitions shows consistency.

Abstract

The fractional integrals and fractional derivatives problem is tackled by using the operator approach. The definition domain E of operators is causal functions.Many properties of fractional integrals are given. Fractional derivatives definition is derived from fractional integrals one. Then an unified definition of fractional integrals and derivatives operator is obtained according to the sign of the real part of the order s. The study utilizes many properties of the Euler's gamma and beta functions and their extensions in R and C. Comparison with the definitions given by other authors (Liouville, Riemann,Liouville-Caputo)is done too.