scholarly journals Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ji Eun Kim ◽  
Su Jin Lim ◽  
Kwang Ho Shon
Keyword(s):  
Exponential Function ◽  
Clifford Analysis ◽  
Regular Function ◽  
Regular Functions ◽  
Quaternion Field ◽  
Hypercomplex Structure ◽  
Cauchy Riemann

We define a new hypercomplex structure ofℝ3and a regular function with values in that structure. From the properties of regular functions, we research the exponential function on the reduced quaternion field and represent the corresponding Cauchy-Riemann equations in hypercomplex structures ofℝ3.

scholarly journals Regular functions with values in a noncommutative algebra using Clifford analysis

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1747-1755
Author(s):  
Su Lim ◽  
Kwang Shon
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Clifford Analysis ◽  
Regular Function ◽  
Complex Variables ◽  
Partial Differential ◽  
Noncommutative Algebra ◽  
Regular Functions ◽  
Commutative Subalgebra ◽  
Pauli Matrices

We construct a noncommutative algebra C(2) that is a subalgebra of the Pauli matrices of M(2;C), and investigate the properties of solutions with values in C(2) of the inhomogeneous Cauchy-Riemann system of partial differential equations with coefficients in the associated Pauli matrices. In addition, we construct a commutative subalgebra C(4) of M(4;C), obtain some properties of biregular functions with values in C(2) on in C2 x C2, define a J-regular function of four complex variables with values in C(4), and examine some properties of J-regular functions of partial differential equations.

scholarly journals Inverse mapping theory on split quaternions in Clifford analysis

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1883-1890 ◽  
Author(s):  
Ji Kim ◽  
Kwang Shon
Keyword(s):  
Clifford Analysis ◽  
Regular Function ◽  
Inverse Mapping ◽  
Split Quaternions ◽  
Cauchy Riemann ◽  
Mapping Theory

We give a split regular function that has a split Cauchy-Riemann system in split quaternions and research properties of split regular mappings with values in S. Also, we investigate properties of an inverse mapping theory with values in split quaternions.

scholarly journals The Regularity of Functions on Dual Split Quaternions in Clifford Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ji Eun Kim ◽  
Kwang Ho Shon
Keyword(s):  
Power Series ◽  
Clifford Analysis ◽  
Differential Operators ◽  
Regular Function ◽  
Split Quaternion ◽  
Split Quaternions ◽  
Cauchy Riemann ◽  
Dual Split Quaternion

This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function onΩ⊂ℂ2×ℂ2that has a dual split Cauchy-Riemann system in dual split quaternions.

scholarly journals The Boundary Value Problem with Haseman-shift for k-regular Functions on Unbounded Domains in Clifford Analysis

2016 ◽  
Vol 8 (1) ◽  
pp. 38
Author(s):  
Yan Zhang
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Unique Solution ◽  
Clifford Analysis ◽  
Boundary Value ◽  
Regular Function ◽  
Unbounded Domains ◽  
Regular Functions

In this paper, we introduce the boundary value problem with Haseman shift for $k$-regular function on unbounded domains, and give the unique solution for this problem by integral equation<br />method and fixed-point theorem.

scholarly journals Linear BVPs and SIEs for Generalized Regular Functions in Clifford Analysis

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Pingrun Li ◽  
Lixia Cao
Keyword(s):  
Clifford Analysis ◽  
Complex Analysis ◽  
Boundary Value ◽  
Regular Function ◽  
Singular Integral Equations ◽  
Value Theory ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Regular Functions ◽  
Plemelj Formula

We study some properties of a regular function in Clifford analysis and generalize Liouville theorem and Plemelj formula with values in Clifford algebra An(R). By means of the classical Riemann boundary value problem and of the theory of a regular function, we discuss some boundary value problems and singular integral equations in Clifford analysis and obtain the explicit solutions and the conditions of solvability. Thus, the results in this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.

PROPERTIES OF REGULAR FUNCTIONS OF A QUATERNION VARIABLE MODIFIED WITH TRI-COMPLEX QUATERNION

2021 ◽  
Vol 10 (5) ◽  
pp. 2663-2673
Author(s):  
Ji Eun Kim
Keyword(s):  
Complex Number ◽  
Regular Function ◽  
Complex Numbers ◽  
Riemann Equation ◽  
Regular Functions ◽  
Complex Quaternion ◽  
Algebraic Properties ◽  
Quaternion Structure ◽  
Cauchy Riemann

In a quaternion structure composed of four real dimensions, we derive a form wherein three complex numbers are combined. Thereafter, we examined whether this form includes the algebraic properties of complex numbers and whether transformations were necessary for its application to the system. In addition, we defined a regular function in quaternions, expressed as a combination of complex numbers. Furthermore, we derived the Cauchy-Riemann equation to investigate the properties of the regular function in the quaternions coupled with the complex number.

Regular functions on dual split quaternions in Clifford analysis

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 17-27
Author(s):  
Ji Kim ◽  
Kwang Shon
Keyword(s):  
Power Series ◽  
Clifford Analysis ◽  
Differential Operators ◽  
Split Quaternions ◽  
Regular Functions ◽  
Cauchy Riemann

This paper shows expressions of a power series for the form of dual split quaternions and provides differential operators in dual split quaternions. The paper also represents a power series of dual split regular functions by using a dual split Cauchy-Riemann system in dual split quaternions.

scholarly journals A Class of Linear Boundary Value Problem for $k$-regular Functions in Clifford Analysis

2016 ◽  
Vol 8 (3) ◽  
pp. 57
Author(s):  
Yan Zhang
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Clifford Analysis ◽  
Boundary Value ◽  
Regular Function ◽  
Integral Equation Method ◽  
Linear Boundary ◽  
Regular Functions ◽  
Linear Boundary Value Problem

In this paper, we introduce the linear boundary value problem for $k$-regular function, and give an unique solution for this problem by integral equation method and fixed-point theorem.

scholarly journals Spherical Coefficients of Slice Regular Functions

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Amedeo Altavilla
Keyword(s):  
Differential Equations ◽  
Regular Function ◽  
Countable Family ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Spherical Expansion ◽  
Derivatives Of

AbstractGiven a quaternionic slice regular function f, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of f with those of its slice derivative $$\partial _{c}f$$ ∂ c f obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.

scholarly journals Some Remarks on Boundary Behavior of Analytic and Meromorphic Functions

1955 ◽  
Vol 9 ◽  
pp. 79-85 ◽  
Author(s):  
F. Bagemihl ◽  
W. Seidel
Keyword(s):  
Meromorphic Function ◽  
Uniqueness Theorem ◽  
Meromorphic Functions ◽  
Boundary Behavior ◽  
Regular Function ◽  
Simple Manner ◽  
Regular Functions ◽  
Almost All ◽  
Jordan Arcs

This paper is concerned with regular and meromorphic functions in |z| < 1 and their behavior near |z| = 1. Among the results obtained are the following. In section 2 we prove the existence of a non-constant meromorphic function that tends to zero at every point of |z| = 1 along almost all chords of |z| < 1 terminating in that point. Section 3 deals with the impossibility of ex tending this result to regular functions. In section 4 it is shown that a regular function can tend to infinity along every member of a set of spirals approach ing |z| = 1 and exhausting |z| < 1 in a simple manner. Finally, in section 5 we prove that this set of spirals cannot be replaced by an exhaustive set of Jordan arcs terminating in points of |z| = 1; Theorem 3 of this section can be interpreted as a uniqueness theorem for meromorphic functions.

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