On fractional derivatives and primitives of periodic functions

TL;DR

This paper proves that fractional derivatives or primitives of periodic functions are generally not periodic, except for the zero function, highlighting fundamental limitations in fractional calculus related to periodicity.

Contribution

It establishes a fundamental non-periodicity result for fractional derivatives and primitives of periodic functions, clarifying limitations in fractional calculus.

Findings

Fractional derivatives of periodic functions are not periodic.

Fractional primitives of periodic functions are not periodic.

The only periodic function with a periodic fractional derivative or primitive is the zero function.

Abstract

In this paper we prove that the fractional derivative or the fractional primitive of a $T$-periodic function cannot be a $\tilde{T}$-periodic function, for any period $\tilde{T}$, with the exception of the zero function.