A fractional Borel-Pompeiu type formula for holomorphic functions of two complex variables

TL;DR

This paper extends fractional calculus to holomorphic functions of two complex variables by deriving a fractional Borel-Pompeiu formula, generalizing classical integral formulas with fractional operators.

Contribution

It introduces a fractional Borel-Pompeiu type formula for holomorphic functions of two complex variables, building on previous work with fractional operators in quaternionic analysis.

Findings

Derived a fractional Borel-Pompeiu formula for holomorphic functions

Extended classical integral formulas to fractional setting

Established new fractional operator calculus in complex variables

Abstract

The present paper is a continuation of our work [11], where we introduced a fractional operator calculus related to a fractional ${\psi}-$Fueter operator in the one-dimensional Riemann-Liouville derivative sense in each direction of the quaternionic structure, that depends on an additional vector of complex parameters with fractional real parts. This allowed us also to study a pair of lower order fractional operators and prove the associated analogues of both Stokes and Borel-Pompieu formulas for holomorphic functions in two complex variables.