Some fractional integral and derivative formulas revisited

Abstract

In the most common literature about fractional calculus, we find that $_{a}D_{t}^{α}f\left( t\right) =\,_{a}I_{t}^{-α}f\left( t\right) $ is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of $_{a}I_{t}^{α}f\left( t\right) $ and $_{a}D_{t}^{α}f\left( t\right) $. In this sense, we prove that $_{0}D_{t}^{α}f\left( t\right) =\,_{0}I_{t}^{-α}f\left( t\right) $ is true for $f\left( t\right) =t^{ν-1}\log t$, and $f\left( t\right) =e^{λt}$, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for $_{-\infty }D_{t}^{α}\left\vert t\right\vert ^{-δ}$ and $_{-\infty }I_{t}^{α}\left\vert t\right\vert ^{-δ}$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that $_{-\infty }D_{t}^{α}\left\vert t\right\vert ^{-δ}=\,_{-\infty }I_{t}^{-α}\left\vert t\right\vert ^{-δ}$ holds true