Some fractional integral and derivative formulas revisited

Abstract

In the most common literature about fractional calculus, we find that ${}_a{D}_t^{\hat{I}\pm }f\left(t\right)={}_a{I}_t^{-\hat{I}\pm }f\left(t\right)$ is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of ${}_a{I}_t^{\hat{I}\pm }f\left(t\right)$ and ${}_a{D}_t^{\hat{I}\pm }f\left(t\right)$. In this sense, we prove that ${}_0{D}_t^{\hat{I}\pm }f\left(t\right)={}_0{I}_t^{-\hat{I}\pm }f\left(t\right)$ is true for $f\left(t\right)=t^{\hat{I}½-1}\log t$, and $f\left(t\right)=e^{\hat{I}»t}$, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for ${}_{-\infty }{D}_t^{\hat{I}\pm }{\left|t\right|}^{-\hat{I}´}$ and ${}_{-\infty }{I}_t^{\hat{I}\pm }{\left|t\right|}^{-\hat{I}´}$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that ${}_{-\infty }{D}_t^{\hat{I}\pm }{\left|t\right|}^{-\hat{I}´}={}_{-\infty }{I}_t^{-\hat{I}\pm }{\left|t\right|}^{-\hat{I}´}$ holds true

17 pages