Generalized Fourier transforms applied to fractional derivatives and Quantum Statistics Distributions

TL;DR

This paper extends Fourier transforms to power-law and quantum statistical functions, enabling new definitions of fractional derivatives and Fourier series for generalized functions and distributions.

Contribution

It introduces a generalized Fourier transform applicable to power-law and quantum distribution functions, facilitating fractional derivatives and series for generalized functions.

Findings

Derived an integral formula using the generalized Fourier transform.

Defined fractional derivatives of distributions and Fourier series for periodic functions.

Extended Fourier analysis tools to quantum statistical distributions.

Abstract

In the present article the author extends the Fourier transform to a more general class of functions; First to power-law functions with integer and half-integer exponents then to the widely used quantum statistics function (Fermi-Dirac and Bose-Einstein-distributions). The results are used to derive rather straightforwardly an integral formula. Moreover, generalized Fourier transforms are used to define fractional derivatives of distributions (generalized functions) and Fourier series (periodic functions) as well.