CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

Plemelj Formula for Cauchy Type Integral on Certain Distinguished Boundary in Universal Clifford Analysis

Under the foundation of Cauchy integral formula on certain distinguished boundary for functions with values in universal Clifford algebra, we define the Cauchy type integral with values in a universal Clifford algebra, obtain its Cauchy principal value and Plemelj formula on certain distinguished boundary.

PLEMELJ FORMULA OF CAUCHY TYPE INTEGRAL ON STEIN MANIFOLDS

Let D be a bounded domain with piecewise C^（1） smooth orientable boundary on Stein manifolds, and let Φ（z） be a Cauchy type integral with Bochner-Martinelli kernel Ω（φ^V, S^-, S） and Holder continuous density function φ（ζ）, the authors define a solid angular coefficient α（t） at the point t∈δD, prove that there exist the interior and outer limit values Φ^±（t） under the meaning of the Cauchy principal value, and obtain the more general Plemelj formula and jump formula.

ON THE ERROR OF QUADRATURE FORMULAE FOR CAUCHY PRINCIPAL VALUE INTEGRALS BASED ON PIECEWISE INTERPOLATION

We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the estimates where quadrature error) is determined for fixed i and which means that not only the. order, but also the coefficient of the main term of is determined. The behaviour of these error constants is compared -with the corresponding ones obtained for the. method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.

BOUNDARY BEHAVIOR OF CAUCHY-TYPE INTEGRALS IN CLIFFORD ANALYSIS

In this article,the authors discussed the boundary behavior for the Cauchy- type integrals with values in a Clifford algebra,obtained some Sochocki–Plemelj formulae and Privalov–Muskhelishvili theorems for the Cauchy-type integral taken over a smooth surface by rather simple method.

Lamb's integral formulas of two-phase saturated medium for soil dynamic with drainage

When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb＇s integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot＇s equation and Betti＇s theorem （the reciprocal theorem）. According to the basic solution to Biot＇s equation, Green＇s function Gij and three terms of Green＇s function G4i, Gi4, and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe＇s solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.

HIGHER ORDER SINGULAR INTEGRAL EQUATIONS ON COMPLEX HYPERSPHERE

A theory of a class of higher order singular integral under the operator （L f） （u） =[u1 σf/σu1（u） - u1σf/σu1（u） ＋ f（u）] is given. We transform the higher order singular integral to a usual Cauchy integral, extend the permutation formula of the higher order singular integral deduced by Qian and Zhong in [4] to a general case, and discuss the regularization problem of the higher order singular integral equations with Cauchy kernel and variable coefficients on complex hypersphere.

ON THE METHOD OF SOLUTION FOR A KIND OFNONLINEAR SINGULAR INTEGRAL EQUATION

The solutions of the nonlinear singular integral equation , t 6 L, are considered, where L is a closed contour in the complex plane, b ≠ 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.

无界域上全纯Cliffordian函数的若干性质及其Cauchy型积分的边值特性

首先介绍了定义于Euclidean空间,取值于实Clifford代数的全纯Cliffordian函数.然后利用正则函数的性质讨论了全纯Cliffordian函数的若干性质和空间性质.主要借用第一类拟置换得出全纯Cliffordian函数的等价条件,研究了它与正则函数之间的关系.接下来以Cauchy积分公式和Plemelj公式为基础,得出了开拓定理.最后定义了无界域上的Cauchy型积分,并证得它在Cauchy主值意义下收敛.同时利用一些重要的积分估值得出了无界域上的Plemelj公式.

Unifying the theory of integration within normal-,Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators （which considers normally ordered, antinormally ordered and Weyl ordered product of operators as its special cases）. The s-ordered operator expansion （denoted by s…s ） formula of density operators is derived, which isρ=2/1-s∫d^2β/π〈-β｜ρ｜β〉sexp{2/s-1（s｜β｜^2-β＊α＋βa-αα）}s The s-parameterized quantization scheme is thus completely established.

Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.

Curve integral with path independent in orthogonal curvilinear coordinate system

It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or three dimensional rectangular coordinate systems. Firstly, according to the coordinate transformation, the condition that the line integral is the path independent in the polar coordinate system is obtained easily from the Green's theorem in two-dimensional rectangular coordinate system and the condition is extended to arbitrary two-dimension orthogonal curvilinear coordinates. Secondly, through the coordinate transformation relationship and the area projection method, the Stokes formula in three-dimensional rectangular coordinate system is promoted to the spherical coordinate system and cylindrical coordinate system, and the condition that the line integral is a path independent is obtained. Furthermore, the condition is extended to arbitrary three-dimension orthogonal curvilinear coordinates. Lastly, the conclusions are made.

Solution of Cauchy's Problem for Wave Equations in Higher Space Dimensions by Means of D'Alembert's Formula

A simple method for solving Cauchy’s problem of wave equations in higher space dimensions with initial condition of separated variables, has been given by using D’Alembert’s formula and some examples have been shown.

Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems

Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

Solution of Singular Integrals in Mathematical Model of Mode I Crack near Strength Mismatched Interface

Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influences the plastic or cohesive zone at the crack tip in parent body. The mathematical model for load line opening of the crack near the interface in linear elastic regime involves singular integrals. The paper presents explicit solution of these integrals with the help of Cauchy’s principal value theorem. Cases of thin and thick welds between the materials are investigated. Solutions of the integrals are well substantiated. Final results are provided in a consolidated form.

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.

超空间中次正则函数的Cauchy积分公式

在本文中,首先给出了超空间中次正则函数(sandwich方程D_(x)fD_(x)=0的解)的一些性质,然后证明了超空间中的Cauchy-Pompeiu公式,最后得到了超空间中的Cauchy积分公式和Cauchy积分定理.

Riemann Boundary Value Problems on the Curve of Parabola

In this paper, we study Riemann boundary value problems on the Curve of Parabola. We characterized the functions which are intergrable on the Curve of Parabola. We also study the asymptotic behaviors of Cauchy-type integral and Cauchy principal value integral on the Curve of Parabola at infinity. At the end, we discuss the Riemann boundary value problems for sectionally holomorphic functions with the Curve of Parabola as their jump curve and obtain the explicit form.

基于q-Bessel多项式求柯西奇异积分方程数值解

提出了基于q-Bessel多项式的离散配置法,并求解柯西奇异积分方程.首先基于q-Bessel多项式对柯西奇异积分方程中未知函数进行逼近,利用平滑变换消除积分项的奇异性.然后将积分方程转换成带有Gaussian-Legendre配置点的代数方程组,并转化成矩阵形式对其进行求解.同时给出了该方法的误差界.最后通过与文献中的其他方法进行比较,验证了该方法的可行性和有效性.

定积分在两点展开的渐近公式

对于具有m+n阶连续导数的被积函数的定积分,通过多次分部积分给出了在积分上限和积分下限两点处同时展开的定积分的渐近展开式,其余项类似于Taylor公式的积分型余项,这种展开式可看作是Taylor公式的一种推广.被积函数在积分上下限处的值有不同情形,展开式也会随之变化而具有多种形式.通过分析与举例也发现这种展开式的近似计算优于Taylor公式的近似计算,而且在某些积分不等式的证明中也体现了其快捷方便的优点.