Definitions of real order integrals and derivatives using operator approach

TL;DR

This paper introduces a unified operator-based framework for defining real order integrals and derivatives, extending classical concepts to fractional and transcendental orders using gamma and beta functions.

Contribution

It provides a novel operator approach to define and analyze real order integrals and derivatives, including fractional and transcendental cases, based on Euler's gamma and beta functions.

Findings

Defined s-order integral operator J^s for positive and fractional s

Derived k-order derivative operator D^k from J^s

Established properties of these operators using gamma and beta functions

Abstract

The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental {\pi} and e). The definition of k-order derivative operator D^k for any positive k (fractional, transcendental {\pi} and e) is derived from the definition of J^s. Some properties of J^s and D^k are given and demonstrated. The method is based on the properties of Euler's gamma and beta functions.