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

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.