A new operator giving integrals and derivatives operators for any order at the same time

TL;DR

The paper introduces a novel operator that simultaneously generalizes integral and derivative operators for any order, encompassing classical, fractional, and complex orders, and applies to causal functions.

Contribution

A new unified operator, called the Raoelinian operator, that generalizes integral and derivative operators for any real or complex order, including fractional cases.

Findings

Recovers classical integral and derivative operators as special cases.

Provides relations for e and π order integrals and derivatives.

Applicable to monomials and potentially to anticausal functions.

Abstract

Let E be the set of integrable and derivable causal functions of x defined on the real interval I from a to infinity, a being real, such f(a) is equal to zero for x lower than or equal to a. We give the expression of one operator that yields the integral operator and derivative operators of the function f at any s-order. For s positive integer real number, we obtain the ordinary s-iterated integral of f. For s negative integer real number we obtain the |s|-order ordinary derivatives of f. Any s positive real or positive real part of s complex number corresponds to s-integral operator of f. Any s negative real number or negative real part of s complex number corresponds to |s|-order derivative operator of f. The results are applied for f being a monom. And remarkable relations concerning the e and {\pi} order integrals and e and {\pi} order derivatives are given, for e and {\pi} transcendental numbers. Similar results may also be obtained for anticausal functions. For particular values of a and s, the operator gives exactly Liouville fractional integral, Riemann fractional integral, Caputo fractional derivative, Liouville-Caputo fractional derivative. Finally, the new operator is neither integral nor derivative operator. It is integral and derivative operators at the same time. It deserves of being named:raoelinian operator is proposed.