Differentiable, Holomorphic, and Analytic Functions on Complex $\Phi$-Algebras

TL;DR

This paper develops an order-theoretic framework for differentiable, holomorphic, and analytic functions on complex $\

Contribution

It introduces an order-theoretic approach to complex $\

Findings

Improved Cauchy-Hadamard formulas for complex vector lattices.

Proved that analytic functions are holomorphic in complex $\

Established a connection between order convergence and complex differentiability.

Abstract

Using the notion of order convergent nets, we develop an order-theoretic approach to differentiable functions on Archimedean complex $\Phi$-algebras. Most notably, we improve the Cauchy-Hadamard formulas for universally complete complex vector lattices given by both authors in a previous paper in order to prove that analytic functions are holomorphic in this abstract setting.