Principal values for the Cauchy integral and rectifiability

We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,

{\varepsilon}} \frac{1}{\xi-z}\, d\mu(\xi),\end{displaymath}">

exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.

     

On the Cauchy Principal Value of the Khenkin-Ramirez Singular Integral in Strictly Pseudoconvex Domains of ℂ <Superscript>n</Superscript>

We show that in the multidimensional case (unlike the complex plane) the Cauchy principal value of the Khenkin-Ramirez singular integral in strictly pseudoconvex domains is equal to the limit value of this integral inside the domain.Original Russian Text Copyright © 2005 Kytmanov A. M. and Myslivets S. G.The first author was supported by a grant of the President of the Russian Federation and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1212.2003.1); the second author was supported by the Krasnoyarsk Region Science Foundation (Grant 12F0063C).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 625–633, May–June, 2005.     

一种Cauchy型主值积分的求积公式的一致收敛性

该文首先给出Cauchy型主值积分φ(wf,x)的一种求积公式φ_m^*(wf,x),然后证明序列$φ_m^*(wf,x)}_m=2^∞在整个闭区间[-1,1]上是一致收敛到Cauchy型主值积分φ(wf,x)的,同时给出它的误差界.     

Uniform approximations to Cauchy principal value integrals of oscillatory functions

This paper presents an interpolatory type integration rule for the numerical evaluation of Cauchy principal value integrals of oscillatory integrands , where -1<τ<1, for a given smooth function f(x). The proposed method is constructed by interpolating f(x) at practical Chebyshev points and subtracting out the singularity. A numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast Fourier transform. The validity of the method has been demonstrated by several numerical experiments.     

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_xfD_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.     

A note on solving Cauchy integral equations of the first kind by iterations

The successive approximation method was applied for the first time by N.I. Ioakimidis to solve practical cases of a Cauchy singular integral equation: the airfoil one. In this paper we study a more general case. We prove the convergence of the method in this general case. The proposed method has been tested for two kernels which are particularly important in practice. Finally, some numerical examples illustrate the accuracy of the method.     

Conditions of convergence of boundary values of Cauchy type integrals

In a domain G bounded by a rectifiable Jordan curve γ let there be given a sequence of analytic functions {f_n(z)} representable by Cauchy-Lebesgue type integrals $$f_n (z) = \smallint _y \frac{{\omega _n (\zeta )}}{{\zeta - z}}d\zeta .$$ A theorem is established which enables one to determine from the convergence in measure of {ω_n(ζ)} on a set e ?γ whether or not there is convergence in measure on the same set of {f_n(ζ)}, the angular boundary values of the functionsf_n(z).     

A Hermitian Cauchy formula on a domain with fractal boundary

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called Dirac operator, which factorizes the Laplacian; monogenic functions may thus also be seen as a generalization of holomorphic functions in the complex plane. Hermitian Clifford analysis offers yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitian (or h-) monogenic functions, of two Hermitian Dirac operators which are invariant under the action of the unitary group. In Brackx et al. (2009) [8] a Clifford-Cauchy integral representation formula for h-monogenic functions has been established in the case of domains with smooth boundary, however the approach followed cannot be extended to the case where the boundary of the considered domain is fractal. At present, we investigate an alternative approach which will enable us to define in this case a Hermitian Cauchy integral over a fractal closed surface, leading to several types of integral representation formulae, including the Cauchy and Borel-Pompeiu representations.     

Modified Gauss rules for approximate calculation of some strongly singular integrals

The approach we follow consists in transforming the numerical evaluation of hyper-singular integrals into the calculation of a nearly singular integral whose mass is distributed according to a positive parameter ε. To evaluate the latter we apply a Gauss quadrature formula associated with a nearly singular weight function. It is estimated the error in terms of ε. Some numerical results are presented.     

Angular limits of the integrals of the Cauchy type

Integrals of the Cauchy type extended over the boundary A of a general compact set A in the complex plane are investigated. Necessary and sufficient conditions on A are established guaranteeing the existence of angular limits of these integrals at a fixed zA for all densities satisfying a Hölder-type condition at z     

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family {In,d,p} of Selberg-like integrals that arise in the computation of the algebraic-geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.     

On the Cauchy Problem for the Equation of Finite-depth Fluids

Some properties of the singular integral operator G(⋅) and the solvability of Cauchy problem for the singular integral-differential equations (1.1) and (1.2) of finite-depth fluids are studied.     

关于一类非线性双曲方程组的二维Cauchy问题

本文通过引进新的广义解定义，对一类非线性双曲方程组的二维Ｃａｕｃｈｙ问题，证明了解的存在唯一性．并且，解可能含δ波．     

第三类典型域上的Cauchy型积分

研究了第三类典型域上的Cauchy型积分，给出了相应的定义，得到了一些结果，使典型域上的Cauchy型积分进一步完善。     

Cauchy Integral Formulae in Rn

In this note p((D)) =(D)m + b1(D)m-1+…+bm is a polynomial Dirac operator in Rn,where (D) =∑nj=1 ej ?(e)/(e)xj is a standard Dirac operator in Rn,bj are the complex constant coefficients.In this note we...     

Cauchy integrals on Lipschitz surfaces in octonionic space

With the newly developed octonion analytic function theory, we confirm the octonionic analogue of the Calderón's conjecture. As application, we obtain the Plemelj formula in octonionic space.