Fractional Integral and Derivative of the 1/r Potential

TL;DR

This paper derives a unified expression for the fractional integral and derivative of the Coulomb potential, enabling new approaches in gravity and electrostatics with fractional calculus.

Contribution

It provides a general formula for fractional derivatives of the 1/r potential applicable across all fractional orders, extending classical potential theory.

Findings

Unified form of fractional derivatives for all orders

Potential applications in modified gravity theories

Framework for fractional charge and mass distributions

Abstract

We calculate the fractional integral and derivative of the potential $1/r$ for all values of the fractional order $-1< \alpha \leq 0$ and $\alpha\geq 0$. We show that the result has the same form for all values of $\alpha$. Applications can be implemented to discuss deformed potential fields resulting from fractional mass or charge densities in gravity and electrostatics problems. The result can also be applied to modify the inverse-quare law gravity as predicted by new physics.