Some fractional integral and derivative formulas revisited

TL;DR

This paper investigates the relationship between fractional derivatives and integrals, proving certain formulas for specific functions and correcting previous inaccuracies in formulas involving functions with infinite limits.

Contribution

It rigorously proves the equivalence of fractional derivatives and integrals for specific functions and corrects existing formulas for functions with infinite limits in fractional calculus.

Findings

Proves $_{0}D_{t}^{eta}f(t) = _{0}I_{t}^{-eta}f(t)$ for specific functions.

Identifies and corrects errors in formulas involving $_{-inite}D_{t}^{eta}$ and $_{-inite}I_{t}^{eta}$ for functions like $|t|^{-inite}$.

Establishes the validity of fractional derivative and integral formulas for functions with infinite limits.

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