On the distributions of boundary values of Cauchy integrals

We use new methods to give short proofs to some known results on the distributions of boundary values of Cauchy integrals. We also indicate some further generalizations.

     

Representation of Cauchy type integrals

The question of obtaining asymptotic and exact representations for a class of Cauchy type integrals is investigated.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 55–63, July, 1969.The author is grateful to F. D. Gakhov for supervising this research.     

On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals

The convergence of a Gauss-Jacobi quadrature rule for the approximate evaluation of Cauchy principal value integrals has been described in recent papers [3] and [4] by the same authors, and will here be proved for Hölder-continuous functions.     

Quadrature processes for integrals of Cauchy type

We study questions relating to convergence of the process $$\int_{ - 1}^{ + 1} \rho (t)\frac{{f(t)}}{{t - x}}dt \approx \sum\nolimits_{k = 1}^n {\alpha _{k,n} } (x)f(x_k^{(n)} )( - 1< x1)$$ wherein the singular integral is taken in the principal value sense. General conditions for convergence in the class of continuously differentiable functionsf are formulated. In the case of the weight function ρ(t)=(√1-t²)^?1, we investigate, under various assumptions onf, the convergence of a specific quadrature process.     

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 convergence of Hunter's quadrature rule for Cauchy principal value integrals

Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf ¹ _–1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp _n generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1].     

Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals

A number of formulae are derived for the numerical evaluation of integrals of the form, whereg(x) possesses one or more simple poles in the interval (–1, 1). The formulae are based on Gauss-Legendre quadrature.     

On the uniform convergence of Cauchy principal values of quasi-interpolating splines

In this paper, quasi-interpolating splines are used to approximate the Cauchy principal value integral $$J(w_{\alpha \beta } f;\lambda ): = \smallint - _{ - 1}^1 w_{\alpha \beta } (x)\frac{{f(x)}}{{x - \lambda }}dx, \lambda \in ( - 1,1)$$ where $w_{\alpha \beta } (x): = (1 - x)^\alpha (1 + x)^\beta ,\alpha ,\beta > - 1.$ . We prove uniform convergence for the quadrature rules proposed here and give an algorithm for the numerical evaluation of these rules.     

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

Numerical evaluation for Cauchy type singular integrals on the interval

The singular integral (SI) with the Cauchy kernel is considered. New quadrature formulas (QFs) based on the modification of discrete vortex method to approximate SI are constructed. Convergence of QFs and error bounds are shown in the classes of functions H^α([−1,1]) and C¹([−1,1]). Numerical examples are shown to validate the QFs constructed.     

Computing some classes of Cauchy type singular integrals with Mathematica software

We constructed an algorithm, [SInt], for computing some classes of Cauchy type singular integrals on the unit circle. The design of [SInt] was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithm. Furthermore, we show how the factorization algorithm described in Conceição et al. (2010) allowed us to construct and implement the [SIntAFact] algorithm for calculating several interesting singular integrals that cannot be computed by [SInt]. All the above techniques were implemented using the symbolic computation capabilities of the computer algebra system Mathematica. The corresponding source code of [SInt] is made available in this paper. Several examples of nontrivial singular integrals computed with both algorithms are presented.     

Vitali type convergence theorems for Banach space valued integrals

Let(Ω,Σ,μ)be a complete probability space and let X be a Banach space.We introduce the notion of scalar equi-convergence in measure which being applied to sequences of Pettis integrable functions generates a new convergence theorem.We also obtain a Vitali type I-convergence theorem for Pettis integrals where I is an ideal on N.     

Pringsheim type tests on the pointwise convergence of integrals conjugate to double fourier integrals

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued function f ∈ L ¹ (?²) with bounded support at a given point (x ₀, g ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier integral of the marginal functions f(x, y ₀), x ∈ ?, and f(x ₀, y), y ∈ ?, at x:= x ₀ and y:= y ₀, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.