A New Operator Giving Integrals and Derivatives Operators at Any Order at the Same Time
Let be the set of integrable and derivable causal functions of defined on the real interval from to infinity, being real, such is equal to zero for lower than or equal to . We give the expression of one operator that yields the integral operator and derivative operators of the function at any -order. For positive integer real number, we obtain the ordinary s-iterated integral of . For negative integer real number we obtain the | |-order ordinary derivatives of . Any positive real or positive real part of complex number corresponds to s-integral operator of . Any negative real number or negative real part of complex number corresponds to | |-order derivative operator of . The results are applied for being a monom. And remarkable relations concerning the and order integrals and and order derivatives are given, for and transcendental numbers. Similar results may also be obtained for anticausal functions. For particular values of and , 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.