Some Fractional Integral and Derivative Formulas Revisited

Abstract

In the most common literature about fractional calculus, we find that Dtαaft=It−αaft is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of Itαaft and Dtαaft. In this sense, we prove that Dt0ft=It−α0ft is true for ft=tν−1logt, and ft=eλt, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for Dtα−∞t−δ and Itα−∞t−δ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that Dtα−∞t−δ=It−α−∞t−δ holds true.