Cauchy Integral Research Articles

NEW GENERALIZATION AND REFINEMENT OF THE DIAMOND-α INTEGRAL CAUCHY–SCHWARTZ INEQUALITY ON TIME SCALES

This paper is concerned with new generalizations and refinements of the diamond-[Formula: see text] integral Cauchy–Schwartz inequality on time scales. The new generalization and refinement of the Cauchy–Schwartz inequality related to diamond integral on time scales are also considered. These obtained inequalities unify continuous inequalities and their corresponding discrete forms associated with Cauchy–Schwartz inequality.

On Cauchy Problem Solution for a Harmonic Function in a Simply Connected Domain with Multi-Component Boundary

Here, we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain with a multi-component boundary. We reduce our investigation to solution of two singular integral equations. If the problem is resolvable, then we restore its solution via the integral Cauchy formula. We present the examples of the solvable problems. The construction involves an auxiliary approximate conformal mapping or the solution of certain singular integral equations.

Ultralow-frequency absorption mechanism of a hybrid membrane resonator with acoustic soft boundary condition

Hybrid membrane resonator (HMR) as a typical metamaterial-based absorber with acoustic hard boundary condition (AHBC) has demonstrated excellent noise absorption abilities, but the challenge lies in achieving broadband absorption of ultralow-frequency noise in the tens of Hz to 150 Hz regime. In this research, we investigate systematically the absorption characteristics, absorption mechanisms, and casual optimality of an HMR with three openings in lower surface of the cavity, which functioned as an acoustic soft boundary condition (ASBC). Compared to the well-studied HMR with AHBC, a new absorption mechanism, which states that most energy dissipates occur in the opening region rather than in the membrane, has been found first. As a result, the HMR with ASBC can demonstrate outstanding ultralow-frequency sound absorption performance with a very small thickness, and the full width at half maximum of the HMR can be enlarged over 7 times. Furthermore, the causal inequalities of the absorbers with ideal AHBC and ASBC are derived based on the Cauchy integral and causality principle. The causal optimality of the proposed absorber is also achieved. This research provides valuable guidelines for the design of excellent ultralow-frequency sound absorbers which could contribute to solving the major issue of noise reduction.

The use of Cauchy-type singular integrals over neighboring intervals to compute induced slip in displaced faults

We present expressions to compute the inverse of a Cauchy-type singular integral equation representing the relation between a double-peaked Coulomb stress in a fault or fracture and the resulting slip gradient in two distinct collinear slip patches. In particular we consider a situation where the patches are close enough to account for the influence of the slip gradient in one patch on the slip-induced shear stress in the other patch and vice versa. This situation can occur during depletion-induced or injection-induced fault slip in subsurface reservoirs for, e.g., natural gas production, hydrogen or CO2 storage, or geothermal operations. The theory for a single slip patch is well-developed but the situation is less clear for a configuration with two patches although the monographs of Muskhelishvili (1953) and Weertman (1996) provide earlier results. We show that the general inverse solution for the coupled two-patch problem requires six auxiliary conditions to ensure six physical requirements: boundedness of the slip gradient at the four end points of the slip patches and vanishing of the integrals of the slip gradient over the patches. Mathematically, the presence of two additional conditions, as compared to earlier formulations, corresponds to two undetermined coefficients in the general solution of the governing integral equation. Numerical simulation confirms that at least one of these is always non-zero in the coupled situation. For a coupled double-patch case with a symmetric pre-slip Coulomb stress pattern, the general inverse solution requires three auxiliary conditions. Moreover the conditions for the asymmetric case may be reduced to a set of four again, but these are different from the sets of four obtained earlier by Muskhelishvili (1953) and Weertman (1996). We illustrate the theory with a numerical example in which the evaluation of the Cauchy integrals is performed with a modified version of augmented Gauss–Chebyshev quadrature that relies on analytical inversion.

Mathematics-Informed Neural Network for Acoustic Scattering by Parallel Plates

This paper proposes a mathematics-informed neural network (MINN) approach for resolving the long-term challenge in wave scattering modeling. The central innovation lies in integrating Cauchy–Riemann equations into machine learning architectures. By incorporating Cauchy integrals and boundary conditions, the neural network successfully learns to numerically produce matrix kernel factorization for Wiener–Hopf analytical models. To validate and demonstrate the approach, a benchmark case of wave scattering from parallel hard–soft plates is studied by comparing the machine learning results with the available analytical solutions. The proposed MINN approach could provide a new route to extensively enhance the theoretical modeling capability for several wave scattering and fluid mechanics problems. The code can be found at https://github.com/lscapku/MINN.

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

Solvability of nonlinear boundary value problems for non-sloping Timoshenko-type isotropic shells of zero principal curvature

We study the solvability of boundary value problem for a system of second order partial differential equations under boundary given conditions describing the equilibrium of elastic non-sloping isotropic inhomogeneous shells with free boundary in the framework of the Timoshenko shear model. The base of the study method are the integral representations for generalized motions involving arbitrary functions, which also involve arbitrary holomorphic functions. The arbitrary functions ate determined so that the generalized motions satisfy a linear system of equations and linear boundary conditions extracted from the original boundary value problem. The holomorphic functions are sought as Cauchy type integrals with real densities. The integral representations allow us to reduce the initial boundary value problem to a nonlinear operator equations for generalized motions in the Sobolev space. While studying the solvability of this operator equation, the most essential point is to invert it with respect to the linear part. As a result, the work is reduced to an equation, the solvability of which is established on the base of the contracting mapping principle.

Riemann boundary value problems on the Archimedean spiral

This paper studies the Riemann boundary value problems on the Archimedean spiral. We characterize the functions which are integrable on the Archimedean spiral. We also study the asymptotic behaviors of Cauchy-type integral and Cauchy principal value integral on Archimedean spiral at the origin and infinity. At the end, we discuss the Riemann boundary value problems for sectionally holomorphic functions with the Archimedean spiral as their jump curve and obtain the singular integral representations of the solutions if solvable.

Quantitative interpretations of potential fields: from parametric to geostructural recalculations

The relevance. Need to development of an applied algorithm for combined interpretation of heterogeneous geophysical data based on unified physical analogy, focused on solving the inverse problem of potential theory. The wave approximation of the cross-section of mining massif fits into the paradigm of dissipative structuring of nonequilibrium media acts as a special case of the above analogy. The task of the study is to adapt this algorithm to recalculations of both potential and non-potential geofields. In the list of nonpotential geofields the remote sensing data and digital terrain model are the most interesting as the data with non-regulated access. Practical significance of the research is associated with the approbation of a family of algorithms in production projects of both geological and engineering nature. The aim. To minimize the family of equivalent solutions to the inverse problem by mutual verification of heterogeneous models of the anomaly-forming source. The objects. Development of the geofields combined interpretation algorithm and recalculation of geofields in depth for tracing elements of stratification and geological fractures. The methods. Formed by software and algorithmic development, which is based on deterministic analytical representations: computation of Cauchy integral, iterative application of direct solution, spectrum estimation. The results. Reduced to the set of parametric cross-sections derived with recalculations with minimum a priori geological information with final geological consideration. The combination of visualizations of analytical continuation, singular points and results of automated fitting admits the reconstruction of geostructural model of parametric cross-section with significant reliability. Extension of standard methods of quantitative interpretation by means of wave approximations includes quasi-seismic halftone visualizations verifying the mentioned reconstructions.

Cauchy type integrals and a boundary value problem in a complex Clifford analysis

Cauchy type integrals and a boundary value problem in a complex Clifford analysis

A Method to Determine Bubble Distribution in Liquid Using Data of Inverse Acoustical Scattering

Abstract Information on bubble distributions in liquids is required for various applications. Employment of inverse acoustic scattering is the usual path to determine these distributions. This path is based on solving a Fredholm first kind integral equation leading to an ill-posed mathematical problem. The usual regularization methods for such a problem are quite complex and require introduction of some tuning parameters. Meanwhile, as shown in this paper, another method works well for media, where acoustic waves propagate with the small losses. This method is based on extraction of a singular Cauchy integral in the above-mentioned equation and of the further inversion of this integral. Such a regularization via inversion is a simple operation that gives numerically stable solutions. Here this regularization is described, verified using the method of manufactured solutions and validated with the well-known already published experimental data.

Compactness of the Family of A(z)-Analytic Functions

In recent years, the Beltrami equation has garnered the attention of numerous researchers for the study of its analytical properties. Therefore, in this paper, we investigate certain properties of an analogue of the Cauchy integral for A(z)-analytic functions, utilizing the analytical properties of the Beltrami equation. Additionally, we obtain the compactness conditions for a family of functions within an A(z)-lemniscate.

A Local Cauchy Integral Formula for Slice-Regular Functions

AbstractWe prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.

Complex Analysis and Residue Theorem

The study of the properties of analytical functions is described as complex analysis. The residue theorem is an important conclusion in complex analysis. This paper introduces the origin of imaginary numbers from Cardano Formula defines the conversion of complex number formats from the Euler Theorem, and proves the intermediate theorem Cauchy Integral Formula before reaching our final conclusion and the goal of the paper, Residue Theorem. The name of the theorem comes from the concept of residue, which is defined using a functions Laurent series. We could then derive the Residue Theorem from the Cauchy Integral Theorem, also called the Cauchy-Goursat Theorem. We will be able to formalize our prior, ad hoc method of computing integrals over contours encompassing singularities. Additionally, it is a theorem that may be applied to zero-pole qualities and curved integral properties. The Residue Theorem is the basis of many essential mathematical facts revolving around line integrals, particularly in solving ODEs and PDEs and describing physics models.

Comparison of the Cauchy Integral Formula and Residue Theorem on the Calculation of Definite Integrals

Integration is a useful mathematical tool which is applied in a wide range of studies and assists people to solve many problems. Despite methods of real analysis, complex integrations in complex analysis are more beneficial and more convenient than real integrations under certain circumstances. Cauchy’s Residue Theorem and Cauchy Integral Formula are two methods used to solve complex integrals. Cauchy’s Residue Theorem helps to solve complex integrals via the calculation involving singular points, whereas Cauchy Integral Formula is implemented to solve complex integrals by finding out poles enclosed by the contour. Herein, the comparison and analysis of the process of solving both real and complex integrations between Cauchy’s Residue Theorem and Cauchy Integral Formula are demonstrated through sample questions involving exponential integrations and polynomial integrations. This passage also includes definitions for Cauchy’s Residue Theorem, Cauchy Integral Formula, corollary of Cauchy Integral Formula, steps and solutions of sample contour integral questions.

BINN: A deep learning approach for computational mechanics problems based on boundary integral equations

We proposed the boundary-integral type neural networks (BINN) for the boundary value problems in computational mechanics. The boundary integral equations are employed to transfer all the unknowns to the boundary, then the unknowns are approximated using neural networks and solved through a training process. The loss function is chosen as the residuals of the boundary integral equations. Regularization techniques are adopted to efficiently evaluate the weakly singular and Cauchy principal integrals in boundary integral equations. Potential problems and elastostatic problems are mainly concerned in this article as a demonstration. The proposed method has several outstanding advantages: First, the dimensions of the original problem are reduced by one, thus the degrees of freedom are greatly reduced. Second, the proposed method does not require any extra treatment to introduce the boundary conditions, since they are naturally considered through the boundary integral equations. Therefore, the method is suitable for complex geometries. Third, BINN is suitable for problems on the infinite or semi-infinite domains. Moreover, BINN can easily handle heterogeneous problems with a single neural network without domain decomposition.

Relationship Between Cauchy Integral Theorem and Residue Theorems

Cauchy integral theorem belongs to an extremely important part of complex functions, which is a fundamental bridge, and people can derive Cauchy integral theorem from the residue theorem. Cauchy's integral theorem is generally applied in many higher mathematics, is an important theorem concerning path integrals of fully pure functions. It claims that if there are two different paths from a point to another point and the function is fully pure between these two different paths, then it can be derived that the two path integrals of the function are equal. It is widely believed that the a generalization of Cauchy integral theorem and Cauchy integral formula is just like the residue theorem, and the flexible use of the residue theorem can easily solve some difficult problems in complex functions. Therefore, this paper will use the definition and derivation process of Cauchy residue theorem and integral theorem to demonstrate the specific connection between them and their practical application.

Representations of generalized Hardy functions in Beurling’s tempered distributions

Let B be a proper open subset in $${{\mathbb {R}}}^N$$ and C be a regular cone in $${{\mathbb {R}}}^N$$ . On our previous paper of Acta Scientiarum Mathematicarum 85, 595–611 (2019), we have defined the space of generalized Hardy functions, $$G_{\omega ^*,A}^p(T^B)$$ , $$1< p \le 2,$$ and $$A \ge 0$$ , and have shown that the functions in $$G_{\omega ^*,A}^p(T^B)$$ have distributional boundary values in the weak topology of Beurling tempered distributions, $${\mathcal {S}}_{(\omega )}^\prime $$ . In this paper we show that if the distributional boundary values are convolutors in Beurling ultradistributions of $$L_2$$ -growth, then the functions in $$G_{\omega ^*,0}^p(T^C)$$ , $$1< p \le 2,$$ can be represented as Cauchy and Poisson integral of the boundary values in $${\mathcal {S}}_{(\omega )}^\prime $$ .

A foundational theory of an extended Cauchy integral for generalized regulated real-valued functions on

In this manuscript, we provide a generalization of the Cauchy integral to appropriate families of real-valued functions defined in R n . The results are motivated by a generalization of regulated functions. The foundational theory for this extended Cauchy integral is developed in detail in this self-contained work. We derive some properties for appropriate families of functions and introduce the Cauchy integral for appropriate step functions. Using these results, the integral for generalized regulated functions is presented next together with some basic properties of the generalized Cauchy integral, including the linearity, the non-negativity and the monotonicity properties. Various illustrative examples on the applicability of our theoretical results are provided.

Cauchy Integral and Pluriharmonic Conjugate Functions on Banach Function Spaces

This paper establishes the boundedness of the Cauchy integral and the pluriharmonic conjugate functions on Banach function spaces. As a special case of our main result, we have the boundedness of the Cauchy integral and pluriharmonic conjugate functions on Morrey spaces.