Some Fractional Integral and Derivative Formulas Revisited

In the most common literature about fractional calculus, we find that D t α a f t = I t − α a f t is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of I t α a f t and D t α a f t . In this sense, we prove that D t 0 f t = I t − α 0 f t is true for f t = t ν − 1 log t , and f t = e λ t , despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for D t α − ∞ t − δ and I t α − ∞ t − δ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that D t α − ∞ t − δ = I t − α − ∞ t − δ holds true.