Some fractional integral and derivative formulas revisited

Abstract

In the most common literature about fractional calculus, we find that $_{a}D_{t}^{\alpha }f\left( t\right) =\,_{a}I_{t}^{-\alpha }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}^{\alpha }f\left( t\right) $ and $_{a}D_{t}^{\alpha }f\left( t\right) $. In this sense, we prove that $_{0}D_{t}^{\alpha }f\left( t\right) =\,_{0}I_{t}^{-\alpha }f\left( t\right) $ is true for $f\left( t\right) =t^{\nu -1}\log t$, and $f\left( t\right) =e^{\lambda t}$, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for $_{-\infty }D_{t}^{\alpha }\left\vert t\right\vert ^{-\delta }$ and $_{-\infty }I_{t}^{\alpha }\left\vert t\right\vert ^{-\delta }$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that $_{-\infty }D_{t}^{\alpha }\left\vert t\right\vert ^{-\delta }=\,_{-\infty }I_{t}^{-\alpha }\left\vert t\right\vert ^{-\delta }$ holds true