A Cauchy kernel for slice regular functions

TL;DR

This paper introduces a new non-commutative Cauchy kernel for slice regular quaternionic functions, providing a novel integral formula, derivative expressions, and several related results in quaternionic analysis.

Contribution

It constructs a regular, non-commutative Cauchy kernel and derives a new Cauchy formula for slice regular functions, advancing quaternionic function theory.

Findings

Established a new Cauchy kernel for slice regular functions

Derived a representation formula for these functions

Expressed derivatives in terms of the Cauchy kernel

Abstract

In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.