Hyperbolic Functions of Bounded Variation and Riemann-Stieltjes Integral involving Strong Partitions of Hyperbolic Intervals

TL;DR

This paper introduces hyperbolic-valued functions of bounded variation and a hyperbolic Riemann-Stieltjes integral using strong partitions of hyperbolic intervals, establishing fundamental relations akin to real analysis.

Contribution

It defines hyperbolic functions of bounded variation and a hyperbolic Riemann-Stieltjes integral via strong partitions, extending classical analysis concepts to hyperbolic numbers.

Findings

Established a hyperbolic analogue of the Riemann-Stieltjes integral.

Proved a fundamental relation between hyperbolic functions of bounded variation and the integral.

Introduced strong partitions as a natural tool for hyperbolic interval analysis.

Abstract

In this paper, we define two types of partitions of an hyperbolic interval: weak and strong. Strong partitions enables us to define, in a natural way, a notion of hyperbolic valued functions of bounded variation and hyperbolic analogue of Riemann-Stieltjes integral. We prove a deep relation between both concepts like it occurs in the context of real analysis.