Some Fractional Integral and Derivative Formulas Revisited

In the most common literature about fractional calculus, we find that Dt alpha aft=It-alpha aft is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of It alpha aft and Dt alpha aft. In this sense, we prove that Dt0ft=It-alpha 0ft is true for ft=t nu-1logt, and ft=e lambda t, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for Dt alpha-infinity t-delta and It alpha-infinity t-delta found in the literature are incorrect; thus, we derive the correct ones, proving in turn that Dt alpha-infinity t-delta=It-alpha-infinity t-delta holds true.