Cauchy Integral Research Articles

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.

STABILITY OF SYSTEMS OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL

Singular Cauchy integral equations have been widely used for mathematical simulation of the actual physical and technical systems. They are considered universal at every level of simulation beginning with quantum field theory and up to strength analysis of the underground constructions. Therefore investigating system stability of such models under perturbation of their absolute terms and coefficients appears an urgent scientific task. The aim of the study is to show various aspects of stability of singular Cauchy integral sets of equations which are generalizing simulation models of the primal problems of the elasticity theory for homogeneous isotropic bodies. The methods of study are based on the properties of the Cauchy singular integral, on the general theory of Fredholm operators. When in use, systems of the singular integral equations are reduced to a set of Fredholm integral equations of the second kind and a set of the boundary value problems for analytic functions. The key results of the study are the following: development of the general determination method of the system index for singular integral equations, proof of the system stability against perturbations of the absolute terms of the set. Against perturbations of the boundary coefficients, the singular integral system is unstable. Demonstration of the stability of the singular integral Cauchy sets generalizing primal problems of the elasticity theory appears a significantly new result. The research of singular integral equations sets has been performed conducted on the space of functions satisfying the Holder condition. However the main research results prove to be true if we operate random functions converting in mean square. Stability of singular integral equations sets against perturbations of the absolute terms lays a foundation for calculus of approximations in real world tasks of defining the built-in stress of an elastic complex body.

Чисельний аналіз деформації в’язкого тіла під діею сил поверхневого натягу

Among the boundary value problems of hydrodynamics of viscous fluid, an important from a practical point of view the class of tasks in the region with a free (previously unknown) boundary, which is determined in the process of direct solving, is distinguished. One of the possible approaches to solving such problems in such an area is a method of hydrodynamic potentials, which translates the basic complexity of research and numerical calculations into certain boundary integral equations, which relate only to the boundary of a given region and take into account the boundary conditions directly. This approach allows you to immediately identify unknown values at the boundary without calculating them in the entire area. This advantage distinguishes the method of boundary integral equations from other methods. The aim of the research was to develop a method for numerical solution of the problems of the viscous fluid movement with a previously unknown boundary. The problem of deformation of a viscous elliptical cylinder under the action of forces of surface tension is formulated and solved in the article. The boundary integral equations are used, which are considered together with the kinematic contour condition. An algorithm for the steps of the numerical solution of the system of boundary integral equations for the problem of viscous fluid with a previously unknown boundary was implemented and computational features were analyzed. On the basis of the conducted research the regularities of the deformation process of the viscous elliptical cylinder under the influence of the forces of surface tension, depending on the viscosity of the liquid and the coefficient of surface tension , were established. A convex contour with continuous curvature deforms into a circle, carrying out fading oscillations around the equilibrium position. Zones of stable synchronous oscillations of points of the contour around the equilibrium position are observed, if , and when , ; in both cases, the density of the liquid was taken equal to one. The liquid body reaches the equilibrium position without oscillation, if . Key words: nucleus, contour discretization interval, curvature, value of the Cauchy integral, surface tension forces, elliptic cylinder

Plane contact problem between a rigid punch and a bidirectional functionally graded medium

Analytical and computational solutions are presented for the sliding frictional contact problem between a bidirectional functionally graded material (BFGM) half-plane and an arbitrarily shaped rigid punch. The plane-strain elasticity is regarded in the solutions. Independent shear modulus variations are imposed through the lateral and the thickness directions of the elastic medium. Fourier transformation techniques are utilized in the derivation of the field quantities. The quartic characteristic equation is solved using the Ferrari's method. A singular integral equation (SIE) of the second kind is then formulated considering a known displacement gradient on the contact surface. Solution of the SIE is performed for the flat and triangular punch profiles via an expansion-collocation method, quadrature integration techniques and a recursive integration method for the Cauchy integral. Finite element analyses (FEA) the same contact problems are also implemented choosing the augmented Lagrange method as the iterative algorithm. The parametric analyses indicate the reliability and validity of both procedures developed in this study. Extra numerical results are generated to show the influences of the problem parameters on the surface stresses and contact forces. It is observed that the contact responses can be influentially controlled upon imposing a bidirectional gradation in an elastic material.

Boundary behavior of Hardy spaces on angular domains

The aim of this paper is to characterize the behavior of boundary limits of functions in Hardy space on angular domains. We investigate that the Cauchy integral of a function $$f\in L^{p}(\partial D_a)$$ is in $$H^{p}(D_a).$$ We also prove that the functions in $$H^{p}(D_a)$$ are the Cauchy integral of their non-tangential boundary limits. In addition, we establish the orthogonality of non-tangential boundary limits of functions in $$H^{p}(D_a)$$ and $$H^{q}(D_a)$$.

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.

On Existence of Solutions of Nonlinear Equilibrium Problems on Shallow Inhomogeneous Anisotropic Shells of the Timoshenko Type

We prove the existence of solutions to the geometrically nonlinear boundary value problem for elastic inhomogeneous anisotropic shallow shells with rigidly clamped edges in the framework of the S.P. Timoshenko shear model. The boundary value problem is reduced to one nonlinear operator equation, whose solvability is established using the principle of compressed maps. The investigation method is based on integral representations for generalized displacements containing holomorphic functions determined by boundary conditions involving the theory of Cauchy type integrals with real densities.

Closed‐form solutions for a circular inhomogeneity in nonlinearly coupled thermoelectric materials

AbstractThermoelectricity is a class of material possessing the ability of interconverting heat and electricity. For the sake of high conversion efficiency, inclusions/fibers are usually introduced into thermoelectric materials. However, the discontinuity of material property and geometry brought by inclusions/fibers might result in severe stress concentration and thus greatly affect the mechanical properties of thermoelectric material in its engineering service. This paper introduces a nonlinear coupled thermal–electrical–mechanical formulation for the plane problem of a circular inhomogeneity embedded in a thermoelectric medium under a combined uniform electric current density and energy flux. A special attention is directed towards the induced thermal stress field which has often neglected in the literature. Using methods based on Cauchy integrals, the complex potentials of the electric field, temperature field and associated stress field are derived explicitly in both the inhomogeneity and the surrounding matrix. The analysis shows that the electrically and thermally induced stress has a linear relationship with the energy flux applied at infinity, but has a nonlinear relationship with the remote electric current density. Numerical simulations are also performed to examine how the inhomogeneity influences the electrically induced thermal stress concentration on the interface. The presented results can be directly used for reliability consideration in design and optimization of thermoelectric composites with circular fibers/inclusions.

Analysis of an orthotropic elastic plane problem of a half plane with an edge crack using a mapping function

It is a challenging problem to derive a solution for an orthotropic elastic plane problem using a mapping function. The complex variable method is employed to introduce a mapping function. Then, comparatively arbitrary configurations can be analyzed. The boundary condition equation is represented by two complex variables. This fact makes solving the present problem difficult. This difficulty is conquered by applying Cauchy integrals. Also, mapping functions represented by an infinite terms of fractional expressions are introduced to conquer the mathematical difficulties of the Cauchy integral in the process of the derivation of the stress function. One of two stress functions is derived by analytical continuation. The final exact stress functions are represented by an irrational mapping function in a closed form. Stress components are represented by one complex variable. Therefore, the calculation of the stress components is easier than that of an isotropic problem. Half planes with a notch or a mound expressed by an irrational mapping function can be solved using their mapping functions. Such a polygonal mapping function can be derived by Schwarz–Christoffel’s transformation. As a demonstration, a half plane with a vertical edge crack subjected to uniform tension is analyzed and the stress distributions are shown for two examples of two characteristic roots of the characteristic equation for the orthotropic elastic plane.

Cauchy Type Integral in Bicomplex Setting and Its Properties

In this paper a bicomplex analogue of a Cauchy type integral in case of a hyperbolic curve of integration is studied. We derive the corresponding Plemelj–Sokhotski formulas for their limit values for all densities satisfying a $${\mathbb {BC}}$$ -Holder-type condition.

Asymptotic error expansions and splitting extrapolation algorithm for two classes of two-dimensional Cauchy principal-value integrals

This paper proposes numerical quadrature rules for two-dimensional Cauchy principal-value integrals of the forms ∫∫Ωf(x,y)(x−s)2+(y−t)2dydx and ∫∫Ωf(x,y)(x−s)(y−t)dydx. The derivation of these quadrature rules is based on the Euler–Maclaurin error expansion of a modified trapezoidal rule for one-dimensional Cauchy singular integrals. The corresponding error estimations are investigated, and the convergence rates O(hm2μ+hn2μ) are obtained for the proposed quadrature rules, where hm and hn are partition sizes in x and y directions, μ is a positive integer determined by integrand. To further improve accuracy, a splitting extrapolation algorithm is developed based on the asymptotic error expansions. Several numerical tests are performed to verify the effectiveness of the proposed methods.

Some properties of a Cauchy type integral in a three-dimensional commutative algebra with one-dimensional radical

In the paper we consider a certain analog of the Cauchy type integral taking values in a three-dimensional commutative algebra over the field of complex numbers with one-dimensional radical. We have established sufficient conditions for the existence of limiting values for such an integral. It is also shown that analogues of Sokhotskii–Plemelj formulas hold.

Support design for permanent mine roadways in tectonically active areas

The authors propose an analytical method of support design for arbitrary cross-section mine roadways subjected to the action of tectonic forces and seismic loads of earthquakes. As distinguished from the existing methods, the new technique allows taking into account influence of a close interface of different rocks. The seismic analysis includes the action of the waves reflected from the interface and the waves refracted through it. The method is based on an analytical solution of 2D elastic problem on stress state of an elastic ring supporting an arbitrary hole in one of two half-infinite media with the common boundary. The solution of the problem has been obtained using the theory of analytical functions of complex variables, Cauchy type integrals, conformal mapping and complex series. The examples of the support design are given.

Real-normalized differentials: limits on stable curves

We study the behaviour of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We describe all possible limits of RN differentials on any stable curve. In particular we prove that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. We further show that the limits of zeros of RN differentials are the divisor of zeros of a twisted differential — an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes. Our main tool is a new method for constructing differentials (in this paper, RN differentials, but the method is more general) on smooth Riemann surfaces, in a plumbing neighbourhood of a given stable curve. To accomplish this, we think of a smooth Riemann surface as the complement of a neighbourhood of the nodes in a stable curve, with boundary circles identified pairwise. Constructing a differential on a smooth surface with prescribed singularities is then reduced to a construction of a suitable normalized holomorphic differential with prescribed ‘jumps’ (mismatches) along the identified circles (seams). We solve this additive analogue of the multiplicative Riemann–Hilbert problem in a new way, by using iteratively the Cauchy integration kernels on the irreducible components of the stable curve, instead of using the Cauchy kernel on the plumbed smooth surface. As the stable curve is fixed, this provides explicit estimates for the differential constructed, and allows a precise degeneration analysis. Bibliography: 22 titles.

The Boundedness of Cauchy Integral Operator on a Domain Having Closed Analytic Boundary

In this paper, we prove that the Cauchy integral operators (or Cauchy transforms) define continuous linear operators on the Smirnov classes for some certain domain with closed analytic boundary.

Computing complex and real tropical curves using monodromy

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.

On Cauchy Integral Theorem for Quaternionic Slice Regular Functions

The aim of this work is to show that the operator G, which has been introduced in Colombo et al. (Trans Am Math Soc 365:303–318, 2013) and whose kernel kerG coincides with the set of quaternionic slice regular functions, is a member of a family of operators with similar properties, such that all the members possess the respective versions of Stokes and Cauchy–type integral theorems. As direct consequences, these theorems are obtained for slice regular functions.

Integral expression for the stationary distribution of reflected Brownian motion in a wedge

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polynomials. To that purpose, we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.

Some remarks on Trefftz type approximations

For the Trefftz method sequences of linearly independent functions satisfying the governing differential equations are needed. For two- and three-dimensional elasticity problems some useful options for obtaining Trefftz trial functions are discussed. The discussion also includes some very useful particular solutions. For three-dimensional elasticity problems four methods are considered: the displacement representation of Papkovich/Neuber, Piltner's complex representation, a hypercomplex displacement representation of Bock, Gürlebeck, Weisz-Patrault, Legatiuk, and H.M. Nguyen, as well as Slobodyanskii’s representation written in real and complex form. For two-dimensional problems the option of using discretized Cauchy integrals is illustrated. Very briefly Trefftz type Radial Basis Functions are mentioned satisfying a homogeneous or inhomogeneous differential equation. This paper is not meant as a complete survey or review on Trefftz methods. The paper presents a collection of personal choices to help future developers of numerical methods based on Trefftz trial functions.