Cauchy Integral Research Articles

On the BMO boundedness of the Cauchy integral on a chord-arc curve

On the BMO boundedness of the Cauchy integral on a chord-arc curve

Cauchy Integral Theorem and its Applications

Italian mathematicians Girolamo Cardano and Raphael Bombelli made the initial dis covery of complex numbers somewhere in the 16th century while attempting to solve a algebra icquestion. The relevance of complex analysis in mathematics, physics, and engineering is incre asing now after hundreds of years of growth, particularly in the areas of algebraic geometry, flu id dynamics, quantum mechanics, and other relatedtopics. This paper discusses three aspects of integration of complex functions, properties and valuation of complex line integral, and Cauchy’s Theorem and its applications. It mainly gives a detailed definition of complex integrals, clarifies the properties of operations such as indefinite integrals and integral paths, and briefly lists several applications of Cauchy’s theorem and proves them. Several theorems are proved from Cauchy’s theorem, Local existence of primitives and Cauchy’s theorem in a disc, and Cauchy’s integral formulas.

Cauchy Integral Operator in Complex Structures with Degeneracies

Cauchy Integral Operator in Complex Structures with Degeneracies

A Derivative Formula for the Solid Cauchy Integral Operator and Its Applications

A Derivative Formula for the Solid Cauchy Integral Operator and Its Applications

On the theory of interpolation of functions on sets of matrices with the Hadamard multiplication

This article is devoted to the problem of interpolation of functions defined on sets of matrices with multiplication in the sense of Hadamard and is mainly an overview. It contains some known information about the Hadamard matrix multiplication and its properties. For functions defined on sets of square and rectangular matrices, various interpolation polynomials of the Lagrange type, containing both the operation of matrix multiplication in the Hadamard sense and the usual matrix product, are given. In the case of analytic functions defined on sets of square matrices with the Hadamard multiplication, some analogues of the Lagrange type trigonometric interpolation formulas are considered. Matrix analogues of splines and the Cauchy integral are given on sets of matrices with the Hadamard multiplication. Some of its applications in the theory of interpolation are considered. Theorems on the convergence of some Lagrange interpolation processes for analytic functions defined on a set of matrices with multiplication in the Hadamard sense are proved. The results obtained are based on the application of some well-known provisions of the theory of interpolation of scalar functions. Data presentation is illustrated by a number of examples.

Double-Preconditioning Techniques for Fractional Partial Differential Equation Solvers

This paper is devoted to the numerical computation of the solution to stationary space fractional Partial Differential Equations (PDEs). To this end, we derive some algorithms for solving fractional algebraic linear systems resulting from the discretization of these space fractional PDEs. The proposed approach is based on an efficient computation of Cauchy’s integrals allowing to estimate positive real powers of a (sparse) matrix A. A first preconditioner M is used to reduce the length of the Cauchy integral contour enclosing the spectrum of MA, hence allowing in some cases for a large reduction of the number of quadrature nodes along the integral contour. Next, ILU-factorizations are used to efficiently solve the linear systems involved in the computation of approximate Cauchy’s integrals. Therefore, the strategy is independent of the way the Cauchy integrals are computed, although in this paper, for the sake of simplicity, we use standard quadrature rules. Based on this approach, we derive some solvers for stationary deterministic or stochastic space fractional Poisson equations which are tested in a series of numerical experiments.

An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on ℝ

This article concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights with finite moments and with smooth external fields with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to w 2. We also study bounds for the first derivatives of a class of functions of the second kind for w 2.

Cauchy Integral and Boundary Value for Vector-Valued Tempered Distributions

Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued analytic functions in tubes TC=Rn+iC obtain vector-valued tempered distributions as boundary values. In a certain vector-valued case, we study the structure of this boundary value, which is shown to be the Fourier transform of the distributional derivative of a vector-valued continuous function of polynomial growth. A set of vector-valued functions used to show the structure of the boundary value is shown to have a one–one and onto relationship with a set of vector-valued distributions, which generalize the Schwartz space DL2′(Rn); the tempered distribution Fourier transform defines the relationship between these two sets. By combining the previously stated results, we obtain a Cauchy integral representation of the vector-valued analytic functions in terms of the boundary value.

Correct Solvability of the Cauchy Problem and Integral Representations of the Solutions for Ultraparabolic Kolmogorov-Type Equations with Two Groups of Spatial Variables of Degeneration

Correct Solvability of the Cauchy Problem and Integral Representations of the Solutions for Ultraparabolic Kolmogorov-Type Equations with Two Groups of Spatial Variables of Degeneration

Numerical algorithms for water waves with background flow over obstacles and topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

Origin of the fractional derivative and fractional non-Markovian continuous-time processes

A complex fractional derivative can be derived by formally extending the integer $k$ in the $k\mathrm{th}$ derivative of a function, computed via Cauchy's integral, to complex $\ensuremath{\alpha}$. This straightforward approach reveals fundamental problems due to inherent nonanalyticity. A consequence is that the complex fractional derivative is not uniquely defined. We explain in detail the anomalies (not closed paths, branch cut jumps) and try to interpret their meaning physically in terms of entropy, friction and deviations from ideal vector fields. Next, we present a class of non-Markovian continuous-time processes by replacing the standard derivative by a Caputo fractional derivative in the classical Chapman-Kolmogorov governing equation of a continuous-time Markov process. The fractional derivative leads to a replacement of the set of exponential base functions by a set of Mittag-Leffler functions, but also creates a complicated dependence structure between states. This fractional non-Markovian process may be applied to generalize the Markovian SIS epidemic process on a contact graph to a more realistic setting.

The precise representative for the gradient of the Riesz potential of a finite measure

Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in \mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.

Interference of internal waves due to two point vortices: linear analytical solution and nonlinear interaction.

In this work, we consider steady two-dimensional interfacial waves in a two-layer stratified fluid, which is induced by a vortex pair located in the lower layer of the fluids. A mathematical model based on the boundary integral equation method and the potential-flow theory is established. The linear analytical solution for the linearized model is given in the form of Cauchy integral and then asymptotic behaviour for large x is presented. The fully nonlinear model is solved by the Jacobian-free Newton–Krylov (JFNK) method numerically. Nonlinear characteristics of wave profiles are identified compared with the linear results under different vortex strengths and the distance between the vortex pair. The amplitude of steady downstream waves is found to vary periodically with respect to the distance of the vortex pair, which can be regarded as the interference between waves produced by each vortex. For equal-strength counter- and co-rotating pairs, the downstream wave heights of linear solutions can be eliminated for some special values of the distance between point vortices, namely, the destructive interference occurs. Meanwhile, the wave only exists between the vortex pair like trapped waves. So does the nonlinear counterpart for counter-rotating pairs, but it could not be diminished with any distance.

BOUNDEDNESS AND COMPACTNESS OF CAUCHY-TYPE INTEGRAL COMMUTATOR ON WEIGHTED MORREY SPACES

AbstractIn this paper we study boundedness and compactness characterizations of the commutators of Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying the minimum regularity condition $C^{2}$ , based on the recent results of Lanzani–Stein and Duong et al. We point out that in this setting the Cauchy type integral is the sum of the essential part which is a Calderón–Zygmund operator and a remainder which is no longer a Calderón–Zygmund operator. We show that the commutator is bounded on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1&lt;p&lt;\infty $ ) if and only if b is in the BMO space on $bD$ . Moreover, the commutator is compact on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1&lt;p&lt;\infty $ ) if and only if b is in the VMO space on $bD$ .

One result on boundedness of the Hilbert transform in Marcinkiewics spaces

In mathematics and in signal theory, the Hilbert transform is an important linear operator that takes a real-valued function and produces another real-valued function. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Also, it is a widely used tool in signal processing. The Cauchy integral is a figurative way to motivate the Hilbert transform. The complex view helps us to relate the Hilbert transform to something more concrete and understandable. Moreover, the Hilbert transform is closely connected with many operators in harmonic analysis such as Laplace and Fourier transforms which have numerous application in partial and ordinary differential equations. In this paper, we study boundedness properties of the classical (singular) Hilbert transform acting on Marcinkiewicz spaces. More precisely, we obtain if and only if condition for boundedness of the Hilbert transform in Marcinkiewicz function spaces.

Symmetrization of a Family of Cauchy-Like Kernels: Global Instability

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its \(L^2\) boundedness. In particular, a famous proof of Melnikov-Verdera (A geometric proof of the \(L^2\) boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7:325–331, 1995) relies upon an iconic symmetrization identity of Melnikov (Analytic capacity: a discrete approach and the curvature of measures. Mat Sb 186(6):57–76, 1995) linking the universal Cauchy kernel \(K_0\) to Menger curvature. Analogous identities hold for the real and the imaginary parts of \(K_0\) as well. Such connections have been immensely productive in the study of singular integral operators and in geometric measure theory. In this article, given any function \(h: {\mathbb {C}} \rightarrow {\mathbb {R}}\), we consider an inhomogeneous variant \(K_h\) of \(K_0\) which is inspired by complex function theory. While an operator with integration kernel \(K_h\) is easily seen to be \(L^2\)-bounded for all h, the symmetrization identities for each of the real and imaginary parts of \(K_h\) show a striking lack of robustness in terms of boundedness and positivity, two properties that were critical in Melnikov and Verdera (A geometric proof of the \(L^2\) boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7:325–331, 1995) and in subsequent works by many authors. Indeed here we show that for any continuous h on \({\mathbb {C}}\), the only member of \(\{K_h\}_h\) whose symmetrization has the right properties is \(K_0\)! This global instability complements our previous investigation (Lanzani and Pramanik, Symmetrization of a Cauchy-like kernel on curves. J Funct Anal (Preprint). arXiv:2001.09375) of symmetrization identities in the restricted setting of a curve, where a sub-family of \(\{K_h\}_h\) displays very different behaviour than its global counterparts considered here. Our methods of proof have some overlap with techniques in recent work of Chousionis and Prat, Some Calderòn–Zygmund kernels and their relation to Wolff capacity. Math Z 282:435–460, 2016) and Chunaev, A new family of singular integral operators whose \(L^2\) boundedness implies rectifiability. J Geom Anal 27:2725–2757, 2017).

Symbolic Computation Applied to Cauchy Type Singular Integrals

The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article.

On Multipoint Padé Approximants whose Poles Accumulate on Contours that Separate the Plane

In this note we consider asymptotics of the multipoint Pade approximants to Cauchy integrals of analytic non-vanishing densities defined on a Jordan arc connecting $ -1 $ and $ 1 $. We allow for the situation where the (symmetric) contour attracting the poles of the approximants does separate the plane.

Cauchy Integral Formula on the Distinguished Boundary with Values in Complex Universal Clifford Algebra

Cauchy Integral Formula on the Distinguished Boundary with Values in Complex Universal Clifford Algebra

Search-Based Path Planning with Homotopy Class Constraints in 3D

Homotopy classes of trajectories, arising due to the presence of obstacles, are defined as sets of trajectories that can be transformed into each other by gradual bending and stretching without colliding with obstacles. The problem of exploring/finding the different homotopy classes in an environment and the problem of finding least-cost paths restricted to a specific homotopy class (or not belonging to certain homotopy classes) arises frequently in such applications as predicting paths for unpredictable entities and deployment of multiple agents for efficient exploration of an environment. In [Bhattacharya, Kumar, Likhachev, AAAI 2010] we have shown how homotopy classes of trajectories on a two-dimensional plane with obstacles can be classified and identified using the Cauchy Integral Theorem and the Residue Theorem from Complex Analysis. In more recent work [Bhattacharya, Likhachev, Kumar, RSS 2011] we extended this representation to three-dimensional spaces by exploiting certain laws from the Theory of Electromagnetism (Biot-Savart law and Ampere's Law) for representing and identifying homotopy classes in three dimensions in an efficient way. Using such a representation, we showed that homotopy class constraints can be seamlessly weaved into graph search techniques for determining optimal path constrained to certain homotopy classes or forbidden from others, as well as for exploring different homotopy classes in an environment. (This is a condensed, non-technical overview of work previously published in the proceedings of Robotics: Science and Systems, 2011 conference [Bhattacharya, Likhachev, Kumar, RSS 2011].)