Slice Regular Functions Research Articles

Cauchy-Riemann operators and local slice analysis over real alternative algebras

We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept of slice-regularity and to the spherical Dirac operator. These results in particular allow to introduce a definition of locally slice-regular function and open the path for local slice analysis. Since Cauchy-Riemann operators factor the corresponding Laplacian operators, the proven formulas let us also obtain several results about the harmonicity and polyharmonicity properties of slice-regular functions.

Slice regular functions in several variables

In this paper, we lay the foundations of the theory of slice regular functions in several (non-commuting) variables ranging in any real alternative ^*-algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function theory is an extension of the classical theory of holomorphic functions of several complex variables. It is based on the construction of a family of commuting complex structures on {mathbb {R}}^{2^n}. One of the relevant aspects of the theory is the validity of a Cauchy-type integral formula and the existence of ordered power series expansions. The theory includes all polynomials and power series with ordered variables and right coefficients in the algebra. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we show that a nonconstant polynomial always has a non empty zero set.

On Fiber Bundles and Quaternionic Slice Regular Functions

The papers (González-Cervantes in Adv Appl Clifford Algebras 31:55, 2021; González-Cervantes Complex Variables and Elliptic Equations 2021) are the first works to apply the theory of fiber bundles in the study of the quaternionic slice regular functions. The main goal of the present work is to extend the results given in González-Cervantes (2021), where the quaternionic right linear space of quaternionic slice regular functions was presented as the base space of a fiber bundle. When the quaternionic right linear space of quaternionic slice regular functions is associated to certain domains then this paper shows that the elements of total space, given in González-Cervantes (2021), are defined from a pair of harmonic functions and a pair of orthogonal vectors. Simplifying the computations presented in González-Cervantes (2021), where each element of the total space is formed by two pair of conjugate harmonic functions and a pair of orthogonal unit vectors. This work also gives some interpretations of the behavior of the zero sets of some quaternionic slice regular polynomials in terms of the theory of fiber bundles.

On Some Quaternionic Generalized Slice Regular Functions

The quaternionic valued functions of a quaternionic variable, often referred to as slice regular functions, has been studied extensively due to the large number of generalized results of the theory of one complex variable. Recently, several global properties of these functions has been found such as a Borel–Pompieu formula and a Stokes’ Theorem from the study of a differential operator, see González-Cervantes and González-Campos (Complex Var Ellipt Equ 66(5):1–10, 2021, https://doi.org/10.1080/17476933.2020.1738410). The aim of this paper is to present a kind of quaternionic generalized slice regular functions that on slices coincide with pairs of complex generalized holomorphic functions associated to Vekua problems. The global properties of these functions are obtained from a perturbed global-type operator and among their local properties presented in this work are the versions of Splitting Lemma, Representation Theorem and a conformal property.

Slice Regular Functions as Covering Maps and Global star -Roots

The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of k-th star -roots of a slice regular function. Moreover, we are also able to compute all the solutions which, quite surprisingly, in the most general case, are in number of k^2. The last part is devoted to compute the monodromy and to present a technique to compute all the k^2 roots starting from one of them.

Slice-by-slice and global smoothness of slice regular and polyanalytic functions

The concept of slice regular function over the real algebra \(\mathbb {H}\) of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let \(\varOmega \subset \mathbb {H}\) be a domain, i.e., a non-empty connected open subset of \(\mathbb {H}=\mathbb {R}^4\). Suppose that \(\varOmega\) intersects \(\mathbb {R}\) and is invariant under rotations of \(\mathbb {H}\) around \(\mathbb {R}\). A function \(f:\varOmega \rightarrow \mathbb {H}\) is slice regular if it is of class \(\mathcal {C}^1\) and, for all complex planes \(\mathbb {C}_I\) spanned by 1 and a quaternionic imaginary unit I (\(\mathbb {C}_I\) is a ‘complex slice’ of \(\mathbb {H}\)), the restriction \(f_I\) of f to \(\varOmega _I=\varOmega \cap \mathbb {C}_I\) satisfies the Cauchy–Riemann equations associated with I, i.e., \(\overline{\partial }_If_I=0\) on \(\varOmega _I\), where \(\overline{\partial }_I=\frac{1}{2}\big (\frac{\partial }{\partial \alpha }+I\frac{\partial }{\partial \beta }\big )\). Given any positive natural number n, a function \(f:\varOmega \rightarrow \mathbb {H}\) is called slice polyanalytic of order n if it is of class \(\mathcal {C}^n\) and \(\overline{\partial }_I^{\,n}f_I=0\) on \(\varOmega _I\) for all I. We define global slice polyanalytic functions of order n as the functions \(f:\varOmega \rightarrow \mathbb {H}\), which admit a decomposition of the form \(f(x)=\sum _{h=0}^{n-1}\overline{x}^hf_h(x)\) for some slice regular functions \(f_0,\ldots ,f_{n-1}\). Global slice polyanalytic functions of any order n are slice polyanalytic of the same order n. The converse is not true: for each \(n\ge 2\), we give examples of slice polyanalytic functions of order n, which are not global. The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations \(\overline{\partial }_I^{\,n}f_I=0\) on \(\varOmega _I\) and as solutions of their global version \({\overline{\vartheta }\,}^nf=0\) on \(\varOmega \setminus \mathbb {R}\). Our quaternionic results extend to the slice monogenic case.

Eigenvalue problems for slice functions

This paper addresses particular eigenvalue problems within the context of two quaternionic function theories. More precisely, we study two concrete classes of quaternionic eigenvalue problems, the first one for the slice derivative operator in the class of quaternionic slice-regular functions and the second one for the Cauchy–Riemann–Fueter operator in the class of axially monogenic functions. The two problems are related to each other by the four-dimensional Laplace operator and Fueter’s Theorem. As an application of a particular case of second order eigenvalue problems, we obtain a representation of axially monogenic solutions for time-harmonic Helmholtz and stationary Klein–Gordon equations.

An interpolation theorem for slice-regular functions with application to very tame sets and slice Fatou–Bieberbach domains in {mathbb {H}}^2

We prove an interpolation theorem for slice-regular quaternionic functions. We define very tame sets in {mathbb {H}}^2 to be the sets which can be mapped by compositions of automorphisms with volume 1 to the set {{mathcal {T}}}=lbrace (2n-1,0), n in {mathbb {N}}rbrace cup lbrace (2n + {mathbb {S}},0), n in {mathbb {N}}rbrace . We then show that any zero set of an entire slice-regular function of one variable embedded in {mathbb {H}}times lbrace 0 rbrace subset {mathbb {H}}^2 is very tame in {mathbb {H}}^2. A notion of slice Fatou–Bieberbach domain in {mathbb {H}}^2 is introduced and, finally, a slice Fatou–Bieberbach domain in {mathbb {H}}^2 avoiding {{mathcal {T}}} is constructed in the last section.

Transcendental operators acting on slice regular functions

Abstract The aim of this paper is to carry out an analysis of five trascendental operators acting on the space of slice regular functions, namely *-exponential, *-sine and *-cosine and their hyperbolic analogues. The first three of them were introduced by Colombo, Sabadini and Struppa and some features of *-exponential were investigated in a previous paper by Altavilla and the author. We show how exp*(f ), sin*(f ), cos*(f ), sinh*(f ) and cosh*(f ) can be written in terms of the real and the vector part of the function f and we examine the relation between cos* and cosh* when the domain Ω is product and when it is slice. In particular we prove that when Ω is slice, then cos*(f ) = cosh*(f * I) holds if and only if f is ℂ I preserving, while in the case Ω is product there is a much larger family of slice regular functions for which the above relation holds.

Quaternionic slice regular functions with some sphere bundles

In the hypercomplex analysis, it is well known that the quaternionic slice regular functions are defined on axially symmetric slice domains and that each quaternion is represented in the form , where and but the mapping allows to see that any axially symmetric slice domain is the base space of a trivial sphere bundle. Remember that a sphere bundle is a useful concept in topology and it is a fiber bundle in which the fibers are spheres of dimension n. The previous idea is the natural motivation of this work whose purpose is to present a definition of the quaternionic slice regular functions on the total space of some sphere bundles, show two non-trivial examples of these function spaces and establish some important results of this function theory.

A New Approach to Slice Analysis Via Slice Topology

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.

Spherical Coefficients of Slice Regular Functions

Given 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 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.

The Product Operator between Bloch-Type Spaces of Slice Regular Functions

The Product Operator between Bloch-Type Spaces of Slice Regular Functions

A Fiber Bundle over the Quaternionic Slice Regular Functions

A Fiber Bundle over the Quaternionic Slice Regular Functions

Weak Slice Regular Functions on the n-Dimensional Quadratic Cone of Octonions

In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called n-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion.

Zeros of slice functions and polynomials over dual quaternions

This work studies the zeros of slice functions over the algebra of dual quaternions and it comprises applications to the problem of factorizing motion polynomials. The class of slice functions over an alternative $*$-algebra $A$ was defined by Ghiloni and Perotti in 2011, extending the class of slice regular functions introduced by Gentili and Struppa in 2006. Both classes strictly include the polynomials over $A$. We focus on the case when $A$ is the algebra of dual quaternions $\mathbb{DH}$. The specific properties of this algebra allow a full characterization of the zero sets, which is not available over general alternative $*$-algebras. This characterization sheds some light on the study of motion polynomials over $\mathbb{DH}$, introduced by Heged\"us, Schicho, and Schr\"ocker in 2013 for their relevance in mechanism science.

A local representation formula for quaternionic slice regular functions

After their introduction in 2006, quaternionic slice regular functions have mostly been studied over domains that are symmetric with respect to the real axis. This choice was motivated by some foundational results published in 2009, such as the Representation Formula for axially symmetric domains. The present work studies slice regular functions over domains that are not axially symmetric, partly correcting the hypotheses of some previously published results. In particular, this work includes a Local Representation Formula valid without the symmetry hypothesis. Moreover, it determines a class of domains, called simple, having the following property: every slice regular function on a simple domain can be uniquely extended to the symmetric completion of its domain.

On the Conformal Mappings and the Global Operator G

Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), Gonzalez Cervantes (Complex Anal Oper Theory 13:2527–2539, 2019) and Gonzalez Cervantes and Gonzalez-Campos (Complex Var Elliptic Equ 65:1–10, 2020, https://doi.org/10.1080/17476933.2020.1738410) [4, 13, 14], respectively. The aim of this work is to show: some relationships between G and the composition operator with the conformal mappings, a conformal covariance property of G along with its interpretations in terms of a covariant functor, all consequences of these facts for the slice regular functions, a Leibnitz rule associated to the operator G and a characterization of the real Components of slice regular functions in terms of a Non-constant Coefficient second order differential equation.

Spectrum and Analytic Functional Calculus for Clifford Operators via Stem Functions

AbstractThe main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and also certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work.

Geometric function theory over quaternionic slice domains

The theory of quaternionic slice regular functions was introduced in 2006 and successfully developed for about a decade over symmetric slice domains, which appeared to be the natural setting for their study. Some recent articles paved the way for a further development of the theory: namely, the study of slice regular functions on slice domains that are not necessarily symmetric. The present work is a panorama of geometric function theory in this new context, where new phenomena appear. For instance, the nature of the zero sets can be drastically different than in the symmetric case. The work includes differential, algebraic, topological properties, as well as integral and series representations, of slice regular functions over slice domains.