Operator of fractional derivative in the complex plane

TL;DR

This paper introduces a fractional derivative operator in the complex plane via Fourier transform, deriving its explicit kernel and analyzing how its uniqueness depends on the derivative's order and the function's poles.

Contribution

It provides an explicit kernel form for the fractional derivative in the complex plane and explores its uniqueness conditions based on derivative order and function poles.

Findings

Explicit kernel of the fractional derivative operator derived

Uniqueness depends on derivative order type and poles of the function

Connections with existing approaches established

Abstract

The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of general derivative operator acting on the functions analytic on a curve in complex plane is deduced and the correspondence with some well known approaches is shown. In particular, it is shown how the uniqueness of the operation depends on the derivative order type (integer, rational, irrational, complex) and the number of poles of considered function in the complex plane.