Quaternion-valued Functions Research Articles

Quaternionic Slice Regular Functions and Quaternionic Laplace Transforms

The functions studied in the paper are the quaternion-valued functions of a quaternionic variable. It is shown that the left slice regular functions and right slice regular functions are related by a particular involution, and that the intrinsic slice regular functions play a central role in the theory of slice regular functions. The relation between left slice regular functions, right slice regular functions and intrinsic slice regular functions is revealed. As an application, the classical Laplace transform is generalized naturally to quaternions in two different ways, which transform a quaternion-valued function of a real variable to a left or right slice regular function. The usual properties of the classical Laplace transforms are generalized to quaternionic Laplace transforms.

Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation

We construct a one-parameter family of generalized Mathieu functions, which are reduced quaternion-valued functions of a pair of real variables lying in an ellipse, and which we call λ-reduced quaternionic Mathieu functions. We prove that the λ-RQM functions, which are in the kernel of the Moisil-Teodorescu operator D+λ (D is the Dirac operator and λ∈R∖{0}), form a complete orthogonal system in the Hilbert space of square-integrable λ-metamonogenic functions with respect to the L2-norm over confocal ellipses. Further, we introduce the zero-boundary λ-RQM-functions, which are λ-RQM functions whose scalar part vanishes on the boundary of the ellipse. The limiting values of the λ-RQM functions as the eccentricity of the ellipse tends to zero are expressed in terms of Bessel functions of the first kind and form a complete orthogonal system for λ-metamonogenic functions with respect to the L2-norm on the unit disk. A connection between the λ-RQM functions and the time-dependent solutions of the imaginary-time wave equation in the elliptical coordinate system is shown.

On (t2, t3)-Zakharov–Shabat equations of generalized Kadomtsev–Petviashvili hierarchies

We review the integration of the Kadomtsev–Petviashvili (KP) hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation, an algebra of functions equipped with a derivation that generalizes the gradient operator, an algebra of quaternion-valued functions, a differential Lie algebra, an algebra of polynomials equipped with the Pincherle differential, and a Moyal algebra. In all these cases, we can formulate and solve the Cauchy problem of the KP hierarchy. In addition, in each of these cases, we derive different Zakharov–Shabat (t2, t3)-equations—that is, different Kadomtsev–Petviashvili equations—and we prove the existence of solutions arising from solutions to the corresponding KP hierarchy.

General weighted class of quaternion-valued functions with lacunary series expansions

General weighted class of quaternion-valued functions with lacunary series expansions

Short-time special affine Fourier transform for quaternion-valued functions

Short-time special affine Fourier transform for quaternion-valued functions

A note on Hardy type inequalities for quaternion-valued functions

A note on Hardy type inequalities for quaternion-valued functions

Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

A generalized monogenic exponential function in ℍ

ABSTRACTThe article presents a monogenic exponential function for the hypercomplex function theory of quaternion-valued monogenic functions in a reduced quaternionic variable, which has essential properties analogous to the complex exponential function , . In particular, it is shown that the Taylor series of the monogenic exponential function based on a complete orthogonal system of monogenic Appell polynomials also generalizes the corresponding complex series representation in a canonical manner.

Some Characterizations for a Quaternion-Valued and Dual Variable Curve

Quaternions, which are found in many fields, have been studied for a long time. The interest in dual quaternions has also increased after real quaternions. Nagaraj and Bharathi developed the basic theories of these studies. The Serret–Frenet Formulae for dual quaternion-valued functions of one real variable are derived. In this paper, by making use of the results of some previous studies, helixes and harmonic curvature concepts in Q D 3 and Q D 4 are considered and a characterization for a dual harmonic curve to be a helix is given.

On the Hausdorff Analyticity for Quaternion-Valued Functions

We consider the concept of the Hausdorff analyticity for functions ranged in real algebras and the corresponding notion of the Hausdorff derivative. Both apply to the real algebra $$\mathbb {H}$$ of Hamilton’s quaternions. The main aim of the work is to compare them with the well-known classes of $$\mathbb {H}$$ -valued functions which have their own definitions of the derivative.

The corresponding Cauchy-Riemann system for dual quaternion-valued functions

The corresponding Cauchy-Riemann system for dual quaternion-valued functions

Quaternion Boundary Element Method for Coupled Exterior and Interior Magnetostatic Fields

In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic problem. The present quaternion-valued BEM is developed by discretizing the quaternion-valued boundary integral equation (BIE). The quaternion-valued BIE can be seen as an extension of the generalized complex variable BIE in 3-D space. In other words, quaternion algebra is an extension of the complex variable in 3-D space. To derive quaternion-valued BIEs, the quaternion-valued Stokes’ theorem is utilized. The quaternion-valued BIEs are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). An analytical scheme is developed to evaluate the CPV by introducing a simple quaternion-valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called “numerical boundary layer” phenomena appear and are remedied by using a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. It is noted that all derivations are done in quaternion algebra. In addition, quaternion algebra is more flexible than vector algebra in solving some 3-D problems from the point view of algebraic space. In deriving BIEs for exterior fields, the conditions at infinity for the quaternion-valued functions are carefully examined. Later, a magnetic sphere in a uniform magnetic field is considered. This problem is a magnetostatic problem of coupled exterior and interior magnetostatic fields. Finally, we apply the present approach to solve the magnetostatic problem. By comparing with exact solutions, the validity of the present approach is checked.

Extreme Points and Some Quaternion Valued Functions in the Unit Ball of $${\mathbb {R}}^3$$ R 3

In this paper, we give a new definition of a class of quaternion function spaces, which will be called \(F(p,\alpha ,s;E)\)-type spaces. Then, we characterize quaternion \(\alpha \)-Bloch functions by functions belonging to \(F(p,\alpha ,s;E)\) spaces under some mild conditions on the weighted function E. Besides, we give some essential properties for the extended little \(\alpha \)-Bloch spaces. Further, we study extreme points on the unit ball of some quaternion function spaces such that the spaces \(F(p,\alpha ,s;E).\)

Q_K-Type Spaces of Quaternion-Valued Functions

In this paper we develop the necessary tools to generalize the Q_K-type function classes to the case of the monogenic functions defined in the unit ball of R^3, some important basic properties of these functions are also considered. Further, we show some relations between Q_K(p,q) and α-Bloch spaces of quaternion-valued functions.

On a three-dimensional Riccati differential equation and its symmetries

A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie point symmetries of the quaternionic Riccati equation are calculated as well as the form of the associated three-dimensional potential of the Schrödinger equation. Using symmetry reductions and relations between the three-dimensional Riccati and the Schrödinger equation, examples are given to obtain solutions of both equations.

Real Paley–Wiener theorem for the quaternion Fourier transform

In this paper we establish the real Paley–Wiener theorems for the quaternion Fourier transform on for quaternion-valued Schwartz functions and -functions, which generalizes the recent results of real Paley–Wiener theorems for scalar- and quaternion-valued -functions.

Berezin transform of slice functions

In a recent paper, we introduced the Berezin transform of quaternionic linear operators and we obtained some properties in the setting of weighted Bergman spaces of slice hyperholomorphic functions. Here, we define the Berezin transform of slice functions, which corresponds to the case of a multiplication operator by certain quaternion-valued slice functions.

Q K Classes in Clifford Analysis

In this paper, we define the classes QK of quaternion-valued functions, then we characterize quaternion Bloch functions by quaternion QK functions in the unit ball of , Further, some important basic properties of these functions are also considered.

Paley-Wiener Theorems for the Two-Sided Quaternion Fourier Transform

In this paper we establish new real Paley-Wiener theorems for the two-sided quaternion Fourier transform. Next, we prove the Roe’s theorem in the context of the quaternion-valued functions. Finally we study the tempered distributions with spectral gaps.

Quaternionic Stockwell transform

ABSTRACTIn this article, we extend the Stockwell transform to the space of all square integrable quaternion-valued functions on the real line, using the convolution of quaternion-valued functions. We prove that the extended Stockwel transform satisfies the Parseval's formula, inversion formula, and uncertainty principle. We also characterize the range of the Stockwell transform on and prove a convolution theorem for the extended Stockwell transform. Applying the convolution theorem, we extend the transform to a suitable Boehmian space of quaternion-valued functions.