Cauchy Integral Formula Research Articles

Numerical Evaluation of Derivatives of Functions

An iterative technique based on the generalized Cauchy integral formula has been developed for the numerical evaluation of the derivatives of a real valued function g(x) such that the function z ↦ g(z)is analytic in a domain which intersects the real axis. The transformed Gauss-Legendre rules meant for the numerical quadrature of analytic function along directed line segments has been employed for the computation of the derivatives. AMS Subject Classification:primary 65D 25

Cauchy Integral Formula

Cauchy-Goursat integral theorem is a pivotal, fundamentally important, and well celebrated result in complex integral calculus. It requires analyticity of the function inside and on the boundary of the simple closed curve. In this study we will investigate the condition(s) under which even though the function is not analytic at a point inside C. Consequently, we will extend the above notion to a finite numbers of points and will present an easy and simple proof of unquestionably the most important, significant and pivotal result known as Cauchy integral formula.

Minimum-resistance shapes in linear continuum mechanics

A necessary optimality condition for the problem of the minimum-resistance shape for a rigid three-dimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint on the volume of the inclusion is obtained through Betti's reciprocal work theorem. It generalizes Pironneau's optimality condition for the minimum-drag shape for a rigid body immersed into a uniform Stokes flow and is specialized for axisymmetric inclusions in axisymmetric and transversal translations. In both cases of translation, the three-dimensional displacement field is represented in terms of generalized analytic functions, and the three-dimensional elastostatics problem is reduced to boundary-integral equations (BIEs) via the generalized Cauchy integral formula. Minimum-resistance shapes are found in the semi-analytical form of functional series from an iterative procedure coupling the optimality condition and the BIEs. They are compared with the minimum-resistance spheroids and with the minimum-resistance spindle-shaped and lens-shaped bodies. Remarkably, in the axisymmetric translation, the minimum-resistance shapes transition from spindle-like shapes to almost prolate spheroidal shapes as the Poisson ratio changes from 1/2 to 0, whereas in the transversal translation, they are close to oblate spheroidal shapes for any Poisson ratio.

Paracomplex Hermitean Clifford Analysis

Substituting the complex structure by the paracomplex structure plays an important role in para-geometry and para-analysis. In this article we shall introduce the paracomplex structure into the realm of Clifford analysis and establish paracomplex Hermitean Clifford analysis by constructing a paracomplex Hermitean Dirac operator \({\mathcal {D}}\) and establishing the corresponding Cauchy integral formula. The theory of paracomplex Hermitean Clifford analysis turns out to be similar to that of complex Hermitean Clifford analysis which recently emerged as a refinement of the theory of several complex variables. It deserves to be pointed out that the introduction of a single operator \({\mathcal {D}}\) in the paracomplex setting has an advantage over the complex setting where complex Hermitean monogenic functions are described by a system of equations instead of being given as null-solution of a single Dirac operator as in the case of classic monogenic functions.

The influence of surface tension upon trapped waves and hydraulic falls

We consider steady two-dimensional free-surface flows past submerged obstructions on the bottom of a channel. The flow is assumed to be irrotational, and the fluid inviscid and incompressible. Both the effects of gravity and surface tension are considered. Critical flow solutions with subcritical flow upstream and supercritical flow downstream are sought using fully nonlinear boundary integral equation techniques based on the Cauchy integral formula. When a second submerged obstruction is included further upstream in the flow configuration in the absence of surface tension, solutions which have a train of waves trapped between the two obstacles before the critical flow have already been found (Dias and Vanden-Broeck 2004 [2]). We extend this work by including the effects of surface tension. Trapped wave solutions are found upstream for small values of the Bond number, for some values of the Froude number. Other types of trapped waves are found for stronger tension when the second obstruction is placed downstream of the hydraulic fall generated by the first obstacle.

Boundary integral equations for 2D thermoelectroelasticity of a half-space with cracks and thin inclusions

Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelectroelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and Stroh orthogonality relations to obtain the integral formulae for the Stroh complex functions, which are piecewise-analytic in the complex half-plane with holes and opened mathematical cuts. Further application of the Stroh formalism allows derivation of the Somigliana type integral formulae and boundary integral equations for a thermoelectroelastic half-space. The kernels of these equations correspond to the fundamental solutions of heat transfer, electroelasticity and thermoelectroelasticity for a half-space. It is shown that the difference between the obtained fundamental solution of thermoelectroelasticity and those presented in literature is due to the fact, that present solution additionally accounts for extended displacement and stress continuity conditions, thus, it is physically correct. Obtained integral equations are introduced into the boundary element approach. Numerical examples validate derived boundary integral equations, show their efficiency and accuracy.

Matrix Cauchy and Hilbert transforms in Hermitian quaternionic Clifford analysis

Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel–Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4 × 4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article.

Some Riemann boundary value problems in Clifford analysis (I)

In this article, we first give the Plemelj formula for functions with parameter by following the classical method. Then by using the higher order Cauchy integral representation formulas, some properties for harmonic functions and bi-harmonic functions are presented, for example, the mean value theorem, the Painlevé theorem, etc. Finally, we consider the Riemann boundary value problems for harmonic functions and bi-harmonic functions in Clifford analysis, the solutions are given in an explicit way.

Normal families of functions for subelliptic operators and the theorems of Montel and Koebe

A classical theorem of Montel states that a family of holomorphic functions on a domain Ω⊆C, uniformly bounded on the compact subsets of Ω, is a normal family. The aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to Lu=0, where L belongs to a wide class of real divergence-form PDOs, comprising sub-Laplacians on Carnot groups, subelliptic Laplacians on arbitrary Lie groups, as well as the Laplace–Beltrami operator on Riemannian manifolds. To this end, we extend another remarkable result, due to Koebe: we characterize the solutions to Lu=0 as fixed points of suitable mean-value operators with non-trivial kernels. A suitable substitute for the Cauchy integral formula is also provided. Finally, the local-boundedness assumption is relaxed, by replacing it with Lloc1-boundedness.

Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties

This paper is concerned with invariant densities for transformations on $\mathbb{R}$ which are the boundary restrictions of inner functions of the upper half plane. G. Letac [9] proved that if the corresponding inner function has a fixed point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ or a periodic point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ with period 2, then a Cauchy distribution $(1/\pi)\mathrm{Im}\left(1/(x-z_{0}) \right)$ is an invariant probability density for the transformation. Using Cauchy's integral formula, we give an easier proof of Letac's result. An easy sufficient condition for such transformations to be isomorphic to piecewise expanding transformations on an finite interval is given by the explicit form of the density. Transformations of the forms $\alpha x + \beta - \sum ^{n }_{k=1}b_{k}/(x-a_{k})$, \:$\alpha x-\sum ^{\infty }_{k=1}\left\{ b_{k}/(x-a_{k})+b_{k}/(x+a_{k}) \right\}$ and $\alpha x +\beta\tan x$ are studied as examples.

Gravity-capillary water waves generated by multiple pressure distributions

Steady two-dimensional free-surface flows subjected to multiple localised pressure distributions are considered. The fluid is bounded below by a rigid bottom, and above by a free-surface, and is assumed to be inviscid and incompressible. The flow is assumed irrotational, and the effects of both gravity and surface tension are taken into account. Forced solitary wave solutions are found numerically, using boundary integral equation techniques, based on Cauchy integral formula. The integrodifferential equations are solved iteratively by Newton's method. The behaviour of the forced waves is determined by the Froude number, the Bond number, and the coefficients of the pressure forcings. Multiple families of solutions are found to exist for particular values of the Froude number; perturbations from a uniform stream, and perturbations from pure solitary waves. Elevation waves are only obtained in the case of a negatively forced pressure distribution.

On the fundamental solution for higher spin Dirac operators

In this paper, we will determine the fundamental solution for the higher spin Dirac operator Qλ, which is a generalisation of the classical Rarita–Schwinger operator to more complicated irreducible (half-integer) representations for the spin group in m dimensions. This will allow us to generalise the Stokes theorem, the Cauchy–Pompeiu theorem and the Cauchy integral formula, which lie at the very heart of the function theory behind arbitrary elliptic higher spin operators.

Color image edge detection based on quaternion and self-organizing map neural network

As the correlation between color components in color image is not considered in the traditional edge detection algorithms,and the detection results will be affected by the threshold value,a new kind of edge detection method for color image was proposed combining quaternion with Self-Organizing Map(SOM).Based on the quaternionic Cauchy integral formula and vector product,the edge characteristic vector was constructed.Then the SOM was trained by the vector and used for edge detection.The experimental results show that the proposed method has stronger retention capacity of the details.

Boundary properties of hypergenic-Cauchy type integrals in Clifford analysis

The aim of this article is to give some boundary properties of hypergenic quasi-Cauchy type integrals in Clifford analysis. First, the Plemelj formula for hypergenic quasi-Cauchy type integrals is given by virtue of the Cauchy integral formula for hypergenic functions. Then, the Privalov theorem for hypergenic quasi-Cauchy type integrals is obtained by virtue of the Plemelj formula.

Matrix Functions of Exponential Order

Both the theoretical and practical investigations of various dynamical systems need to extend the definitions of various functions defined on the real axis to the set of matrices. To this end one uses mainly three methods which are based on 1) the Jordan canonical forms, 2) the polynomial interpolation, and 3) the Cauchy integral formula. All these methods give the same result, say g(A), when they are applicable to given function g(t) and matrix A. But, unfortunately, each of them puts certain restrictions on g(t) and/or A, and needs tedious computations to find explicit exact expressions when the eigen-values of A are not simple. The aim of the present paper is to give an alternate method which is more logical, simple and applicable to all functions (continuous or discontinuous) of exponential order. It is based on the two-sided Laplace transform and analytical continuation concepts, and gives the result as a linear combination of certain n matrices determined only through A. Here n stands for the order of A. The coefficients taking place in the combination in question are given through the analytical continuation of g(t) (and its derivatives if A has multiple eigen-values) to the set of eigen-values of A (numerical computation of inverse transforms is not needed). Some illustrative examples show the effectiveness of the method.

Numerical computation of Cauchy principal value integrals using complex variable techniques

A technique based on the Cauchy integral formula of the theory of analytic functions of a complex variable has been formulated for the numerical evaluation of both real and complex Cauchy value integrals along directed line segments. The numerical verification of the method has been performed by considering some model examples. Mathematics Subject Classification: 65D30

Numerical evaluation of fractional integrals and derivatives of analytic function

A numerical technique based on the Cauchy integral formula of complex analysis has been employed for evaluating the integrals and derivatives of fractional orders of an analytic function. The method of subtraction of singularity has been used to evaluate the contour integrals for a desired degree of accuracy.

The Cauchy–Pompeiu integral formula in elliptic complex numbers

The aim of this article is to give a generalization of the Cauchy–Pompeiu integral formula for functions valued in parameter-depending elliptic algebras with structure polynomial X 2 + βX + α, where α and β are real numbers. As a consequence, a Cauchy integral representation formula is obtained for a generalized class of holomorphic functions.

Cauchy integral formula for generalized analytic functions in hydrodynamics

It is shown that for several classes of generalized analytic functions arising in linearized equations of hydrodynamics and magnetohydrodynamics, the Cauchy integral formulae follow from the one for generalized holomorphic vectors in a uniform fashion. If hydrodynamic fields (velocity, pressure and vorticity) admit representations in terms of corresponding generalized analytic functions, those representations and the Cauchy integral formulae form two essential parts of the generalized analytic function approach, which readily yields either closed-form solutions or boundary integral equations. This approach is demonstrated for problems of axisymmetric and asymmetric Stokes flows, two-phase axisymmetric Stokes flows, two-dimensional and axisymmetric Oseen flows.

Fundamentals of Bicomplex Pseudoanalytic Function Theory: Cauchy Integral Formulas, Negative Formal Powers and Schrödinger Equations with Complex Coefficients

The study of the Dirac system and second-order elliptic equations with complex-valued coefficients on the plane naturally leads to bicomplex Vekua-type equations (Campos et al. in Adv Appl Clifford Algebras, 2012; Castaneda et al. in J Phys A Math Gen 38:9207–9219, 2005; Kravchenko in J Phys A Math Gen 39:12407–12425, 2006). To the difference of complex pseudoanalytic (or generalized analytic) functions (Bers in Theory of pseudo-analytic functions. New York University, New York, 1952; Vekua in Generalized analytic functions. Nauka, Moscow (in Russian); English translation Oxford, 1962. Pergamon Press, Oxford, 1959) the theory of bicomplex pseudoanalytic functions has not been developed. Such basic facts as, e.g., the similarity principle or the Liouville theorem in general are no longer available due to the presence of zero divisors in the algebra of bicomplex numbers. In the present work we develop a theory of bicomplex pseudoanalytic formal powers analogous to the developed by Bers (Theory of pseudo-analytic functions. New York University, 1952) and especially that of negative formal powers. Combining the approaches of Bers and Vekua with some additional ideas we obtain the Cauchy integral formula in the bicomplex setting. In the classical complex situation this formula was obtained under the assumption that the involved Cauchy kernel is global, a very restrictive condition taking into account possible practical applications, especially when the equation itself is not defined on the whole plane. We show that the Cauchy integral formula remains valid with the Cauchy kernel from a wider class called here the reproducing Cauchy kernels. We give a complete characterization of this class. To our best knowledge these results are new even for complex Vekua equations. We establish that reproducing Cauchy kernels can be used to obtain a full set of negative formal powers for the corresponding bicomplex Vekua equation and present an algorithm which allows one their construction. Bicomplex Vekua equations of a special form called main Vekua equations are closely related to stationary Schrodinger equations with complex-valued potentials. We use this relation to establish useful connections between the reproducing Cauchy kernels and the fundamental solutions for the Schrodinger operators which allow one to construct the Cauchy kernel when the fundamental solution is known and vice versa. Moreover, using these results we construct the fundamental solutions for the Darboux transformed Schrodinger operators.