Cauchy-Riemann Research Articles

Cauchy Formulae and Hardy Spaces in Discrete Octonionic Analysis

In this paper, we continue the development of a fundament of discrete octonionic analysis that is associated to the discrete first order Cauchy–Riemann operator acting on octonions. In particular, we establish a discrete octonionic version of the Borel–Pompeiu formula and of Cauchy’s integral formula. The latter then is exploited to introduce a discrete monogenic octonionic Cauchy transform. This tool in hand allows us to introduce discrete octonionic Hardy spaces for upper and lower half-space together with Plemelj projection formulae.

Nonlinear Beltrami equation: lower estimates of Schwarz lemma’s type

Abstract We study a nonlinear Beltrami equation $f_\theta =\sigma \,|f_r|^m f_r$ in polar coordinates $(r,\theta ),$ which becomes the classical Cauchy–Riemann system under $m=0$ and $\sigma =ir.$ Using the isoperimetric technique, various lower estimates for $|f(z)|/|z|, f(0)=0,$ as $z\to 0,$ are derived under appropriate integral conditions on complex/directional dilatations. The sharpness of the above bounds is illustrated by several examples.

A new HEAP-inspired method for embedded power flows and path-related holomorphic properties

Power flow problems can be robustly solved by holomorphic embedding (HE) methods which require the holomorphic property of solution functions near a reference state and also along a path bridging this state and the target state, for determining the effectiveness of series expansion and physical realizability (PRL), respectively, where PRL means the complex conjugacy of solution values at the target state. Existing studies generally assume holomorphic properties beforehand or assert them from a convergence of series expansion or implicit function theorem at separate points, however, which can hardly be applied to confirm the holomorphic property along a path and then PRL (because a path has infinitely many points and cannot be numerically inspected point-by-point). To this end, a new approach PHVACR is enlightened by HEAP and employs an expansion in the rectangular form and also Cauchy–Riemann (CR) conditions to numerically verify holomorphic properties around a path in an arc-by-arc manner. Here, ‘HE’ is short for holomorphic embedding, ‘AP’ in HEAP indicates an arc-length parametrization, and ‘PHVA’ in PHVACR refers to the path-related holomorphy validation with ‘CR’ representing the Cauchy–Riemann conditions. To explain the importance of PHVACR and its performance, counter-intuitive examples are discussed to explain the limitations of existing approaches, and large problems with 10,000+ buses are also simulated to examine the effectiveness and scalability of PHVACR which are compared with other HE methods.

Properties of a polyanalytic functional calculus on the S‐spectrum

AbstractThe Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in , denoted by . This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of . Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

IDENTIFICATION OF SEPARATION CURVES OF STRONGLY CONTRASTING ENVIRONMENTS USING COMPLEX ANALYSIS METHODS

When modeling mass transfer processes (for example, filtration) in porous media, there may be cases of the existence of highly permeable layers, which are separated from the corresponding studied parts by some curves that must be found (identified) in the process of solving the problem. When constructing a mathematical model of the corresponding physical process, we will consider a highly permeable medium as "ideally (theoretically infinitely) permeable." In this case, the desired curve can be considered an equipotential line. This work considers the stationary process of fluid movement in a homogeneous horizontal layer of infinitely large dimensions - a soil massif, which is limited by infinite sections of curves, in particular by the desired curve of theoretical water resistance and the horizontal axis, on which the local speed of movement is known. On the basis of the methods of conformal reflections and total images, an approach to the identification of the curve of the separation of media is proposed. The constructed algorithm is modified for solving nonlinear inverse boundary value problems on quasi-conformal mappings of curvilinear polygonal regions bounded by uncertain streamlines and equipotential lines. The feature of the method proposed is that the summary representation formulae allow to explicitly write the solution to the localized linear (main) part of the obtained system of equations. The unknown coefficients are determined here by solving a non-linear system generated solely by the boundary conditions and numerical analogues of the Cauchy–Riemann equations.

BGG Sequences with Weak Regularity and Applications

We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n}$$\\end{document}. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.

An Atomic Representation for Hardy Classes of Solutions to Nonhomogeneous Cauchy–Riemann Equations

An Atomic Representation for Hardy Classes of Solutions to Nonhomogeneous Cauchy–Riemann Equations

Developing Geometric Reasoning of the Relationship of the Cauchy Riemann Equations and Differentiation

Developing Geometric Reasoning of the Relationship of the Cauchy Riemann Equations and Differentiation

Pre-twisted calculus and differential equations

In this paper we introduce the φA-differentiability for functions f:U⊂Rk→Rn, where U is an open set, A is the linear space Rn endowed with a unital associative commutative algebra product, and φ:U⊂Rk→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.

The Fueter-Sce mapping and the Clifford–Appell polynomials

AbstractThe Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in ${\mathbb{R}}^{n+1}$. This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in $ \mathbb{R}^{n+1}$. The second one is a suitable power of the Laplace operator in n + 1 variables. Another way to get axially monogenic functions is the generalized Cauchy–Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions $\Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$, where $x \in \mathbb{R}^{n+1}$. The expressions obtained is related to a well-known class of Clifford–Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension, we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford–Appell–Fock space and the Clifford–Appell–Hardy space. Finally, using the polyanalytic Fueter-Sce theorems, we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford–Appell polynomials.

On a Generalization of Monge–Ampère Equations and Monge–Ampère Systems

In the present paper, we discuss Monge-Ampère equations from the viewpoint of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space and that its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we show that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equations.

Reconstruction of the Cauchy–Riemann Operator by Complex Integration Operators along Circles

Reconstruction of the Cauchy–Riemann Operator by Complex Integration Operators along Circles

Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields

Abstract We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra 𝔸 {\mathbb{A}} for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to 𝔸 {\mathbb{A}} , which is also known as the Lorch derivative with respect to 𝔸 {\mathbb{A}} , and recently simply called 𝔸 {\mathbb{A}} -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity.

Hörmander’s L^2-Method, {overline{partial }}-Problem and Polyanalytic Function Theory in One Complex Variable

In this paper we consider the classical {overline{partial }}-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the {overline{partial }}-problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.

The additional proof and application of Cauchy-Riemann Equation

As far back as the seventeenth century, Newton and Leibniz introduced and refined the idea of calculus, and the concepts of derivative and differentiation gradually entered the mathematical world as powerful tools. However, after the appearance of the theory of complex functions in the 18th century, the conventional methods used to analyze the properties of complex functions were somewhat limited. Thus, mathematicians Cauchy and Riemann came up with the idea of the “C-R equation” and provided the necessary and sufficient conditions in derivable complex functions based on the research done by d'Alembert. This equation laid the foundation for the analytic function. This paper introduces the idea of the limit of two variables function and shows both the concept of and graphical meaning partial derivative, paving way for the concept of complex function and that of C-R equation. In addition, the paper specifically proves the C-R equation by a much more rigorous method with the routes from the linear equation and applies this important equation to determine the number of and the locations of the differentiable points in a complex plane. It is also one of the factors to determine the analyticity of complex functions. At the end of the paper, an example will be provided to illustrate the application of the C-R equation.

A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator inR4\mathbb {R}^4, denoted byD\mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operatorΔ\Deltain four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e.Δ=D¯D\Delta = \mathcal {\overline {D}} \mathcal {D}to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions ofD2\mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on theSS-spectrum.

On a Spectral Problem for the Cauchy–Riemann Operator with Boundary Conditions of the Bitsadze–Samarskii Type

On a Spectral Problem for the Cauchy–Riemann Operator with Boundary Conditions of the Bitsadze–Samarskii Type

Some Fundamental Results from Complex Analysis

This paper is going to introduce the most basic theory of analytic functions of one complex variable. It begins at the discussion of meaning of complex number and the historical development from the formula of cubic equation to the square root of negative number. In the middle section, which is divided in to four small parts. First part states the expression of complex number z, algebraic properties, and the relationship of each single complex number with whole complex plane. Second part concerns about several elementary functions of complex number. Next part relates to the derivative of complex number, such as the partial derivative and Cauchy-Riemann Equation to verdict that whether a function is analytic in the domain. Last part is the integral of complex functions, which concludes some important theorems with the proof, and examples. After the section of introduction of complex number, it comes to the final section which is talking about an identity involving complex number and the prove of it.

Schwarz Boundary Value Problems for Polyanalytic Equation in a Sector Ring

In this article, we first give a modified Schwarz–Pompeiu formula in a general sector ring with angle \(\vartheta =\frac{\pi }{\alpha },\ \alpha \ge 1/2\) by proper conformal mappings, and obtain the solution of the Schwarz problem for the Cauchy–Riemann equation in explicit forms. Furthermore, we construct some integral operators and discuss their properties in detail. Finally, by virtue of these operators, Schwarz problems for an inhomogeneous polyanalytic equation and for a generalized polyanalytic equation are investigated, respectively.

On Hyperholomorphic Bergman Type Spaces in Domains of $$\mathbb C^2$$

Quaternionic analysis is a branch of classical analysis referring to different generalizations of the Cauchy-Riemann equations to the quaternion skew field \(\mathbb H\) context. In this work we deals with \(\mathbb H-\)valued \((\theta , u)-\)hyperholomorphic functions related to elements of the kernel of the Helmholtz operator with a parameter \(u \in \mathbb H\), just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions. Given a domain \(\Omega \subset \mathbb H\cong \mathbb C^2\), our main goal us to discuss the Bergman spaces theory for this class of functions as elements of the kernel of \({}^{\theta }_{u}\mathcal D [f]= {}^{\theta } \mathcal D [f] + u f\) with \(u\in \mathbb H\) defined in \(C^1(\Omega , \mathbb H)\), where $$\begin{aligned} {}^\theta \mathcal D:= \frac{\partial }{\partial \bar{z}_1} + ie^{i\theta }\frac{\partial }{\partial z_2}j = \frac{\partial }{\partial \bar{z}_1} + ie^{i\theta }j\frac{\partial }{\partial \bar{z}_2}, \theta \in [0,2\pi ). \end{aligned}$$Using as a guiding fact that \((\theta , u)-\)hyperholomorphic functions includes, as a proper subset, all complex valued holomorphic functions of two complex variables we obtain some assertions for the theory of Bergman spaces and Bergman operators in domains of \(\mathbb C^2\), in particular, existence of a reproducing kernel, its projection and their covariant and invariant properties of certain objects.