Cauchy Integral Formula Research Articles

Integral Representation Formulas Related to the Lamé—Navier System

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lame—Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel—Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.

Slice Dirac operator over octonions

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative O(1) case to the non-commutative O(3) case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

New exotic capillary free-surface flows

Abstract

Differential Topological Aspects in Octonionic Monogenic Function Theory

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 can use this tool to prove a generalized octonionic Rouché’s theorem by a homotopic argument. As an application we set up a generalized version of Hurwitz theorem which is also a novelty even for the Clifford analysis case.

Frequency domain identification using adaptive Fourier decomposition method with polynomials

The adaptive Fourier decomposition method is an approximation technique of generalised Fourier series, it leads to fast approximations by selecting basis functions in a maximal selection criterion. In recent research studies, it has been efficiently applied in the identification of linear time-invariant systems and a new algorithm is named two-step (T-S) algorithm. In this work, some further modification is made for the T-S algorithm. The improvement is made at the first step, where polynomials are used instead of cauchy integral formula. By doing this, the algorithm becomes simpler and easier to realise. The approximation errors are analysed. Owing to the analysed results, this new T-S algorithm is only to get poles for the finite rational orthogonal basis functions but not the approximation to the systems, the coefficients are estimated by using least-squares methods. The effectiveness is examined through numerical examples that show it takes much less running times and can get comparable approximating results. Besides, the case that errors are included in the frequencies is also studied, the obtained results imply that minor errors in the frequencies would not affect the estimation.

The Cauchy Integral Formula in Hermitian, Quaternionic and $$\mathfrak {osp}(4|2)$$ Clifford Analysis

As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a cornerstone 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 to how hermitian Clifford analysis in euclidean space $${\mathbb {R}}^{2n}$$ of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on $${\mathbb {R}}^{2n}$$ , quaternionic Clifford analysis arose as a further refinement by introducing a so-called hypercomplex structure $${\mathbb {Q}}$$ , i.e. three complex structures ( $${\mathbb {I}}$$ , $${\mathbb {J}}$$ , $${\mathbb {K}}$$ ) which follow the quaternionic multiplication rules, on $${\mathbb {R}}^{4p}$$ , the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Cauchy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy integral formulae for $$\mathfrak {osp}(4|2)$$ -monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.

Dirac Operators with Gradient Potentials and Related Monogenic Functions

We investigate some properties of solutions to Dirac operators with gradient potentials. Solutions to Dirac operators with gradient potentials are called monogenic functions with respect to the potential functions. We establish a Borel–Pompeiu formula, and obtain Cauchy integral formula and mean value formula about such functions. Based on integral formulas, we prove some geometric properties of monogenic functions with respect to the potential functions and construct some integral transforms. The boundedness of integral transforms in Holder space is given. We prove the Plemelj–Sokhotski formula and the Painleve theorem. As applications, we firstly prove that a kind of Riemann–Hilbert problem for monogenic functions with respect to the potential functions is solvable. Explicit representation formula of the solution is also given. We then establish solvability conditions of a kind of general Riemann–Hilbert problem for monogenic functions with respect to the potential functions. Finally, we give some examples about the above Riemann–Hilbert problem.

Integration of Heterogeneous Complex Variable Function and Combination Formula

In this paper, we discuss the integration of heterogeneous complex variable function f(zk)=1 (zk-a)n in curve Cj of heterogeneous complex plane Hkby using heterogeneous Cauchy integral theorem, Cauchy integral formula and residue theory. And then obtain two identical equation of combinatorial mathematics by using two different calculation methods.

Quantum algorithm for matrix functions by Cauchy's integral formula

For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{\im A t}. However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime, where \epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, \e^{\im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.

Generalized tensor function via the tensor singular value decomposition based on the T-product

In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) based on the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors, from which the projection operators and Moore-Penrose inverse of tensors are obtained. We establish the Cauchy integral formula for tensors by using the partial isometry tensors and apply it into the solution of tensor equations. Then we establish the generalized tensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are listed. We define the tensor bilinear and sesquilinear forms and propose theorems on structures preserved by generalized tensor functions. For complex tensors, we established an isomorphism between complex tensors and real tensors. In the last part of our paper, we find that the block circulant operator establishes an isomorphism between tensors and matrices. This isomorphism is used to prove the F-stochastic structure is invariant under generalized tensor functions. The concept of invariant tensor cones is raised.

ДРОБНАЯ ПРОИЗВОДНАЯ КОШИ

In this paper, we introduce a new sort of fractional derivative. For this, we consider the Cauchy's integral formula for derivatives and modify it by using Laplace transform. So, we obtain the fractional derivative formula F(α)(s) = L{(–1)(α)L–1{F(s)}}. Also, we find a relation between Weyl's fractional derivative and the formula above. Finally, we give some examples for fractional derivative of some elementary functions.

The Cauchy Integral Formula for Biregular Function in Octonionic Analysis

In this paper, we mainly study the Cauchy integral formula and mean value theorem for biregular function in octonionic analysis. Octonion is the extension of complex number to non-commutative and non-associative space. Because of the non-associative properties of multiplication, octonion plays an important role in wave equation, Yang-Mills equations, operator theory and so on. In recent years, octonion has become a hot topic for scholars at home and abroad and got many rich results, such as Fourier transform, Bergman kernel, Taylor series and its applications in quantum mechanics. On the basis of two Stokes theorems, we get Cauchy integral formula for biregular function in octonionic analysis by using the methods in dealing with the Cauchy integral formula for biregular function in Clifford analysis and regular function in octonionic analysis. As a direct result we also get the mean value theorem for biregular function in octonionic analysis. This will generalize the corresponding conclusion in complex analysis and Clifford analysis, and lays a solid foundation for the application of octonionic analysis in physics.

The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

The Cauchy integral formula expresses the value of a function f(z), which is analytic in a simply connected domain D, at any point z0 interior to a simple closed contour C situated in D in terms of the values of on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of f(z). At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of f(z) is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle |z| = r and of some circles |z-a| = 1/r for which the formula is valid.

Mixed Boundary Value Problem for an Anisotropic Thermoelastic Half-Space Containing Thin Inhomogeneities

Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and the extended Stroh formalism which allows writing the general solution of thermoelastic problems in terms of certain analytic functions. In addition, with the help of it, it is possible to convert the volume integrals included in the equation into contour integrals, which, in turn, will allow the use of the method of boundary elements. For modelling of solids with thin inhomogeneities, a coupling principle for continua of different dimensions is used. Applying the theory of complex variable functions, in particular, Cauchy integral formula and Sokhotski–Plemelj formula, the Somigliana type boundary integral equations are constructed for thermoelastic anisotropic half-space. The obtained integral equations are introduced into the modified boundary element method. A numerical analysis of the influence of boundary conditions on the half-space boundary and relative rigidity of the thin inhomogeneity on the intensity of stresses at the inclusions is carried out.

An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model

We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion of arbitrary long-range hopping amplitudes, providing a simple framework to investigate arbitrary adiabatic deformations that preserve the chiral symmetry upon the bulk energy bands with any arbitrary winding numbers. Using only elementary techniques of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically rigorous and physically transparent proof of the bulk-boundary correspondence for the generalized SSH model. The multiplicity of robust zero-energy edge modes is shown to be identical to the winding number. On the other hand, nonzero-energy edge modes, if any, are shown to be unstable under adiabatic deformations and not related to the topological invariant. Furthermore, under deformations of small spatial disorder, the zero-energy edge modes remain robust.

K-holomorphic functions in spaces of several complex variables

In this paper, we define k-holomorphic functions in and study their properties. We obtain some conclusions parallel to the properties of holomorphic functions, such as Cauchy integral theorem, Cauchy integral formula of k-holomorphic functions in and mean value theorem. k-holomorphic functions are expansions of holomorphic functions. The properties of k-holomorphic functions are not only wider used in other fields of mathematics but also are important tools to study other subjects, such as elastic dynamics, hydrodynamics and physics.

Hyperbolic Function Theory in the Skew-Field of Quaternions

We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called alpha -hyperbolic harmonic, that are harmonic with respect to the Riemannian metric dsα2=dx02+dx12+dx22+dx32x3α\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} ds_{\\alpha }^{2}=\\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{\\alpha }} \\end{aligned}$$\\end{document}in the upper half space {mathbb {R}}_{+}^{4}={left( x_{0},x_{1},x_{2} ,x_{3}right) in {mathbb {R}}^{4}:x_{3}>0}. If alpha =2, the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function x^{m},(min {mathbb {Z}}), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental alpha -hyperbolic harmonic functions, depending only on the hyperbolic distance and x_{3}, we verify a Cauchy type integral formula for conjugate gradient of alpha -hyperbolic harmonic functions. We also compare these results with the properties of paravector valued alpha -hypermonogenic in the Clifford algebra {{,mathrm{{mathcal {C}}ell },}}_{0,3}.

Split Octonionic Cauchy Integral Formula

Cauchy integral formulas for the quaternions have been known since 1926. This result has previously been extended to the octonions and split quaternions. These techniques are extended to the split octonions. Thus Cauchy integral formulas are now known for all composition algebras over the real numbers.

Non-tangential limits for analytic Lipschitz functions

Let U be a bounded open subset of the complex plane. Let $$0<\alpha <1$$ and let $$A_{\alpha }(U)$$ denote the space of functions that satisfy a Lipschitz condition with exponent $$\alpha $$ on the complex plane, are analytic on U and are such that for each $$\epsilon >0$$ , there exists $$\delta >0$$ such that for all z, $$w \in U$$ , $$|f(z)-f(w)| \le \epsilon |z-w|^{\alpha }$$ whenever $$|z-w| < \delta $$ . We show that if a boundary point $$x_0$$ for U admits a bounded point derivation for $$A_{\alpha }(U)$$ and U has an interior cone at $$x_0$$ then one can evaluate the bounded point derivation by taking a limit of a difference quotient over a non-tangential ray to $$x_0$$ . Notably our proofs are constructive in the sense that they make explicit use of the Cauchy integral formula.

Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

In this paper, we establish higher order Borel-Pompeiu formulas for conformally invariant fermionic operators in higher spin theory, which is the theory of functions on m-dimensional Euclidean space taking values in arbitrary irreducible representations of the Spin group. As applications, we provide higher order Cauchy's integral formulas for those fermionic operators. This paper continues the work of building up basic integral formulas for conformally invariant differential operators in higher spin theory.