Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

K-解析函数的Riemann边值问题

引入了(分片)K-解析函数和Cauchy型K-积分的概念.利用K-对称变换的方法研究了Cauchy型K-积分的某些性质,然后借助函数在曲线上的指标与这些Cauchy型K-积分的性质,得到了K-解析函数类中的Riemann边值问题的可解条件和解的表达式以及它们与指标之间的关系.而解析函数和共轭解析函数都是K-解析函数的特例,文中所得结果,推广了解析函数和共轭解析函数中的相应结论.     

一类具一阶奇性核的奇异积分方程

当(?)是复平面C上的光滑封闭曲线,k（z）是在(?)所围成的有界闭区域上连续.在其内部解析的函数时.借助于奇异积分算子的广义逆.讨论了具一阶奇性核的正则型奇异积分方程:　在H类中的求解问题.作为应用,作者给出了当k（z）是一类有理函数时的具体解法,从而统一并推广了 Cauchy核和Hilbert核奇异积分方程的经典结果.     

多复变数的Cauchy型积分(Ⅰ)超球的Cauchy型积分

<正> §1.1.引言Cauchy 型积分在复变函数论中的重要性不必多说了,它不但有着函数论本身的重要意义,而且是奇异积分方程、边界值问题等学科中不可缺少的工具.但是对于多复变数Cauchy 型积分的研究是不多的.陆启铿与锺同德在[3]中研究了由 Bochner 定义的Cauchy 核所生成的 Cauchy 型积分,并得出了相应的(?)定理.这时候 Cauchy 型积分所定义的函数一般来说并不是解析函数,积分是在区域的整个边界上进行(?)     

正实轴上的Riemann边值问题

本文研究正实轴上的Riemann边值问题.首先,引入沿正实轴剖开的复平面上的全纯函数在无穷远点和原点处主部及阶的概念,相比于经典意义下,这个概念更为广泛.其次,讨论了正实轴上Cauchy型积分和Cauchy主值积分在无穷远点和原点处的性质.基于此,以正实轴为跳跃曲线的分区全纯函数的Riemann边值问题得以详细解决.这个过程有别于经典意义下有限曲线上的Riemann边值问题,且比整个实轴上的Riemann边值问题更为复杂.最后,作为例子讨论了一类矩阵值函数的边值问题,该问题对于正实轴上正交多项式的渐近分析有重要意义.     

Cauchy 型积分的一种边值定理及其应用

对于平面上逐段光滑的曲线,本文证明了 Cauchy 型积分的一种边值定理.它包含经典的 Plemelj-定理作为特例,因而是经典定理的一种推广.文中还构造了 D 函数族,它包含 H 族为真子族,并把边值定理应用于其上,得到了完全类似于经典的关于 H 族的结果,从而可以同样展开学派关于奇异     

双解析函数的Haseman边值问题

本文利用双解析函数的Cauchy型积分和带位移的奇异积分方程方法,研究并得到了双解析函数的Haseman边值问题的一般解的表示式和可解条件以及线性无关解的个数与指标之间的关系.     

积分型Cauchy中值函数若干分析性质

给出"积分型Cauchy中值函数"的定义,对"积分型Cauchy中值函数"的分析性质进行了系统讨论,证明了"积分型Cauchy中值函数"的单调性、可积性、连续性、可微性等分析性质.作为"积分型Cauchy中值函数"的特例,给出了"第一积分中值函数"的定义及"第一积分中值函数"相应的分析性质.     

Clifford分析中无界域上Isotonic函数的Plemelj公式

定义了无界域上Isotonic函数的Cauchy型积分和Cauchy主值积分,考虑了其边界性质,得到了无界域上Isotonic函数的Plemelj公式.     

乘积空间上沿复合曲线的奇异积分与Marcinkiewicz积分(英文)

本文致力于研究混合齐性空间上沿复合曲线的多参数奇异积分和Marcinkiewicz积分.所考虑的复合曲线包含了R~m×R~n(m,n≥2)中的许多经典的曲线.在积分核满足L log~+L型最优尺寸条件下,这些算子的L~P有界性被建立.这些结果很大程度上推广了已有的结果.     

Analytical solution of polynomial equations with an application to the Quintic equation

A recently proposed method for the derivation of exact analytical integral formulae for the zeros of analytic functions (based on the simple discontinuity problem for a sectionally analytic function along the real axis) is applied here to the case of polynomials. The peculiarity of the present application is that the integrals appearing in the closed-form formulae for the sought zeros are interpreted as Cauchy-type principal-value integrals or even as finite-part integrals. The case of the quintic equation with real coefficients is considered in some detail, and it is shown that the roots of this equation can always be obtained in closed form. Numerical results for this equation are also presented. Equations of higher degree can also be solved in closed form under appropriate conditions.     

On estimate for the modulus of continuity of the Cauchy-type integral having a Lipschitz-continuous density

The Cauchy-type integral having a Lipschitz-continuous density is under investigation. It is considered in a Jordan region that has a piecewise regular boundary without cusps and therefore it can be continuously extended over the closure of the region. Then the boundary values form a function, whose modulus of continuity is to be estimated. One parameter-dependent estimate is obtained and one algorithm for its evaluating is developed. The algorithm is demonstrated on some examples.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations

Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhotskii-Plemelj theorem for it as well as a necessary and sufficient condition for the possibility of extending a given Hölder function from such a surface up to a solution of the Moisil-Theodoresco system of partial differential equations in a domain. A formula for the square of a singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between the theory of the Moisil-Theodoresco system of partial differential equations and some versions of quaternionic analysis.     

Some properties of singular integral operators over an open curve

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471048)     

Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof. Lu Jianke

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.     

Cauchy-type integrals of algebraic functions

We consider Cauchy-type integrals