Bicomplex polyharmonicity and polyholomorphy

Normal Families of Bicomplex Holomorphic Functions

In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice $\bold{j}=0$.

Bicomplex algebra and function theory

This treatise investigates holomorphic functions defined on the space of bicomplex numbers introduced by Segre. The theory of these functions is associated with Fueter's theory of regular, quaternionic functions. The algebras of quaternions and bicomplex numbers are developed by making use of so-called complex pairs. Special attention is paid to singular bicomplex numbers that lack an inverse. The elementary bicomplex functions are defined and their properties studied. The derivative of a bicomplex function is defined as the limit of a fraction with nonsingular denominator. The existence of th…

Normal Families of Bicomplex Meromorphic Functions

In the present paper, we introduced the extended bicomplex plane $\bar{\mathbb{T}}$, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about the convergence of the sequences of bicomplex meromorphic functions. Hence the concept of the normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the Fundamental Normality Tests for families of bicomplex holomorphic functions and bico…

双复 Miller-Ross 函数的性质与应用

Properties and applications of the Bicomplex Miller-Ross function

本文引入了双复 Miller-Ross 函数，建立了其递推关系、积分表示、微分关系与双复全纯性等性质，并推导了广义分数阶动力学方程双复扩展的解。

关于具有双复变量的 Bessel 函数与 n 维 Hankel 变换及相干态

On the Bessel function and $n$-dimensional Hankel transform with Bicomplex arguments and coherent states

本文引入双复 Bessel 函数，系统研究其收敛域、递推、积分、微分、全纯性与渐近性质，并据此定义 n 维双复 Hankel 变换、求解偏微分方程及构造新型相干态。

Beltrami方程解的双复变Hardy类与Schwarz边值问题

Bicomplex Hardy Classes of Solutions to Beltrami Equations and the Schwarz Boundary Value Problem

该研究定义了复单位圆盘上双复变Beltrami方程及相关方程解的双复值Hardy类，证明其具有经典全纯Hardy空间的边界行为，推广了已有结果并证明相关Schwarz、Dirichlet边值问题可解且给出解式。