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Integration of Bicomplex Valued Function along Hyperbolic Curve

Chinmay Ghosh, Soumen Mondal

TL;DR

The paper extends line integral theory to bicomplex-valued functions along hyperbolic ($\mathbb{D}$) paths by introducing product-type bicomplex functions, $\mathbb{D}$-bounded variation, and rectifiable $\mathbb{D}$-paths, and by defining a Riemann–Stieltjes integral with respect to $\Gamma$. It proves a bicomplex Fundamental Theorem of Calculus for line integrals using a product-type primitive $F$, and derives properties such as linearity, decomposition via the idempotent basis, and zero integral on closed paths. These results generalize classical complex-analytic line integration to the commutative, zero-divisor setting of bicomplex/hyperbolic algebras and furnish calculus tools for hypercomplex analysis on $\mathbb{BC}$.

Abstract

In this paper, we have defined bicomplex valued functions of bounded variations and rectifiable hyperbolic path. We have studied the integration of product-type bicomplex functions over rectifiable hyperbolic path. Also we have established bicomplex analogue of the Fundamental Theorem of Calculus for line integral.

Integration of Bicomplex Valued Function along Hyperbolic Curve

TL;DR

The paper extends line integral theory to bicomplex-valued functions along hyperbolic () paths by introducing product-type bicomplex functions, -bounded variation, and rectifiable -paths, and by defining a Riemann–Stieltjes integral with respect to . It proves a bicomplex Fundamental Theorem of Calculus for line integrals using a product-type primitive , and derives properties such as linearity, decomposition via the idempotent basis, and zero integral on closed paths. These results generalize classical complex-analytic line integration to the commutative, zero-divisor setting of bicomplex/hyperbolic algebras and furnish calculus tools for hypercomplex analysis on .

Abstract

In this paper, we have defined bicomplex valued functions of bounded variations and rectifiable hyperbolic path. We have studied the integration of product-type bicomplex functions over rectifiable hyperbolic path. Also we have established bicomplex analogue of the Fundamental Theorem of Calculus for line integral.
Paper Structure (3 sections, 11 theorems, 90 equations)

This paper contains 3 sections, 11 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Main Results

Key Result

Proposition 1

Let $\Gamma :\left[ \alpha ,\beta \right] _{\mathbb{D}}\rightarrow \mathbb{BC}$ be of $\mathbb{D}-$bounded variation.Then $\left( a\right)$ If $P$ and $Q$ are partitions of $\left[ \alpha ,\beta \right] _{\mathbb{D}}$ and $P\subset Q$ then $\left( b\right)$ If $\Lambda :\left[ \alpha ,\beta \right] _{\mathbb{D}}\rightarrow \mathbb{BC}$ is also of $\mathbb{D}-$bounded variation and $a,b\in \mathbb

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 26 more