Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.e. continuous and differentiable in the sense of Gateaux) functions of the variable $\sum_{j=1}^k x_j\,e_j$, where $x_1,x_2,\ldots,x_k$ are real, and we prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula in $k$-dimensional ($2\leq k\leq 2n$) real subset of the algebra $\mathbb{A}_n^m$. The present article is generalized of the author's paper [1], where mentioned results are obtained for $k=3$.
In this paper, we mainly consider the Riemann boundary value problems for lower dimensional non-commutative Clifford algebras valued monogenic functions. The solutions are given in an explicit way and concrete examples are presented to illustrate the results.
We consider an arbitrary finite-dimensional commutative associative algebra, $\mathbb{A}_n^m$, with unit over the field of complex number with $m$ idempotents. Let $e_1=1,e_2,e_3$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.e. continuous and differentiable in the sense of Gateaux) functions of the variable $xe_1+ye_2+ze_3$, where $x,y,z$ are real. For mentioned monogenic function we prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula.
Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.~e. continuous and differentiable in the sense of Gateaux) functions of the variable $\sum_{j=1}^k x_j\,e_j$, where $x_1,x_2,\ldots,x_k$ are real, and obtain a constructive description of all mentioned functions by means of holomorphic functions of complex variables. It follows from this d…
In the paper [1] considered a new class of quaternionic mappings, so-called $G$-monogenic mappings. In this paper we prove analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, and the Cauchy integral formula for $G$-monogenic mappings.
The aim of this work is to weaken the conditions of monogenity for functions that take values in one concrete three-dimensional commutative algebras over the field of complex numbers. The monogenity of the function understood as a combination of its continuity with the existence of a Gato derivative.
For $G$-monogenic mappings taking values in the algebra of complex quaternion we prove a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies on the boundary of a domain.
We introduce the Area Integral for octonion-valued monogenic functions in the half-space. It is used to prove the existence of the non-tangential boundary values almost everywhere and of the normal boundary values at a given boundary point for these classes of functions. It is also proved that the non-tangential limit for the scalar component of a monogenic function and those limits for all of its other seven components exist or not exist simultaneously. It turns out that non-commutativity and non-associativity of octonions in certain problems can be circumvented by making use of P. Stein meth…
As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a corner stone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis in euclidean space R^{2n} of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on R^{2n}, quaternionic Clifford analysis arose as a further refinement by introducing a so--called hypercomplex structure Q…
The Fueter-Sce-Qian (FSQ for short) mapping theorem is a two-steps procedure to extend holomorphic functions of one complex variable to slice monogenic functions and to monogenic functions. Using the Cauchy formula of slice monogenic functions the FSQ-theorem admits an integral representation for $n$ odd. In this paper we show that the relation $ Δ_{n+1}^{(n-1)/2}S_L^{-1}=\mathcal{F}^L_n $ between the slice monogenic Cauchy kernel $S_L^{-1}$ and the F-kernel $\mathcal{F}^L_n$, that appear in the integral form of the FSQ-theorem for $n$ odd, holds also in the case we consider the fractional pow…
In this paper we study two-sided (left and right) axially symmetric solutions of a generalized Cauchy-Riemann operator. We present three methods to obtain special solutions: via the Cauchy-Kowalevski extension theorem, via plane wave integrals and Funk-Hecke's formula and via primitivation. Each of these methods is effective enough to generate all the polynomial solutions.
The Radon transform between monogenic and generalized slice monogenic functions
在[J. Bures，R. Lavicka，V. Soucek，Elements of quaternionic analysis and Radon transform，Textos de Matematica 42，Departamento de Matematica，Universidade de Coimbra，2009]，作者描述了依赖于参数的全纯函数与使用Radon和对偶Radon变换在R^（n+1）上定义的单演函数之间的联系。本文的主要目的是进一步发展这种方法。实际上，值在Clifford代数R_n中的函数的Radon变换是广义Cauchy-Riemann方程的映射解，即，单演函数映射到R_n中的参数全纯函数族，类似地，对偶Radon变换也是将上述参数全纯函数族映射到单演函数。本文所考虑的参数全纯函数族可以看作是所谓切片单演函数的推广。本文所解决的问题的一个重要部分是找到一个合适的定义函数空间作为域和目标的积分变换。
In this paper we introduce and study a Bargmann-Radon transform on the real monogenic Bargmann module. This transform is defined as the projection of the real Bargmann module on the closed submodule of monogenic functions spanned by the monogenic plane waves. We prove that this projection can be written in integral form in terms the so-called Bargmann-Radon kernel. Moreover, we have a characterization formula for the Bargmann-Radon transform of a function in the real Bargmann module in terms of its complex extension and then its restriction to the nullcone in $\mathbb C^m$. We also show that t…
A new integral formula for the inverse Fueter mapping theorem
本文给出了构造$k$次轴单演函数的Fueter本原的另一种方法，它是[F.科伦坡岛Sabadini和F. Sommen，The inverse Fueter mapping theorem in integral form using spherical monogenics，Israel Journal of Mathematics，2012].作为副产品，我们得到一个明确的描述Fueter映射的内核。我们还应用我们的方法来获得单位球上的奇异柯西核的Fueter原语。
In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and $a$-points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative Clifford analysis setting so far. We also prove an argument principle for octonionic monogenic functions for isolated zeroes and for non-isolated compact zero sets. In the isolated case we c…
The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function $\tilde f:L_n\to \C$ on the Lie ball $L_n$ in $\C^n$ with its monogenic counterpart $f:B_1(0)\to \C^{n+1}$ via the formula $\tilde f(z) = \int_{S^n}G_\om(z)\bs n(\om)f(\om)\,dμ(\om)$, $z\in L_n.$ The inverse map $\tilde f\mapsto f$ is constructed here using the Cauchy-Hua formula for the Lie ball following the work of M. Morimoto \cite{Mori2}.
Integral formulas and Hodge decomposition in the theory of generalized partial-slice mo-nogenic functions
School of Mathematical Sciences, Anhui University（安徽大学数学科学学院）；Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院纯数学中心）
In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on some sequences of commutative associative algebras over the field of complex numbers. To achieve this goal, we first study the solutions of the so-called characteristic equation on a given sequence of algebras. Further, we investigate monogenic functions on the sequence of algebras and study their relation with solutions of partial deferential equations. The p…
Recently the construction of various integral transforms for slice monogenic functions has gained a lot of attention. In line with these developments, the article at hand introduces the slice Fourier transform. In the first part, the kernel function of this integral transform is constructed using the Mehler formula. An explicit expression for the integral transform is obtained and allows for the study of its properties. In the second part, two kinds of corresponding convolutions are examined: Mustard convolutions and convolutions based on generalised translation operators. The paper finishes b…
We study in detail the one-variable local theory of functions holomorphic over a finite-dimensional commutative associative unital $\mathbb{C}$-algebra $\mathcal{A}$, showing that it shares a multitude of features with the classical one-variable Complex Analysis, including the validity of the Jacobian conjecture for $\mathcal{A}$-holomorphic regular maps and a generalized Homological Cauchy's Integral Formula. In fact, in doing so we replace $\mathcal{A}$ by a morphism $φ: \mathcal{A} \to \mathcal{B}$ in the category of finite-dimensional commutative associative unital $\mathbb{C}$-algebras in…
The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collabo…
