Some fractional integral and derivative formulas revisited

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 ${}_{-\mathrm{\infty }}{D}_t^{\alpha }{\mid t\mid }^{-\delta }$ and ${}_{-\mathrm{\infty }}{I}_t^{\alpha }{\mid t\mid }^{-\delta }$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that ${}_{-\mathrm{\infty }}{D}_t^{\alpha }{\mid t\mid }^{-\delta }={}_{-\mathrm{\infty }}{I}_t^{-\alpha }{\mid t\mid }^{-\delta }$ holds true