Cauchy Integral Theorem Research Articles

The Cauchy integral formula for (p, q)-monogenic functions with α-weight in superspace

First, some properties for -monogenic functions with α-weight in superspace are obtained. Then the Cauchy–Pompeiu formula in superspace is given. Finally, the Cauchy integral formula and the Cauchy integral theorem for -monogenic functions with α-weight in superspace are proved.

Application of Cauchy’s Residue Theorem in Several Improper Integrals

Calculating definite integrals in complex functions requires the Cauchy's residue theorem, which is a key concept in the complex variables. It is based on several ideas, including the isolated singular points theory, the Laurent theorem, and the Cauchy integral theorem. Nevertheless, it is highly challenging to calculate closed curves using the conventional integrals method since numerous formulas must be substituted. The singular points must all be isolated in the case of a holomorphic function that belongs to an enclosing contour and is positively oriented, at which time the contour integral can be used to resolve the issue. Additionally, the problem can be solved by using visual aids to help people understand them. By doing so, the notion of solving the problem will be made clearer and some calculating steps will be cut short. This study exhibits the residue theorem's potent efficiency in computing improper integrals and other difficult problems.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

Pre-twisted calculus and differential equations

In this paper we introduce the φA-differentiability for functions f:U⊂Rk→Rn, where U is an open set, A is the linear space Rn endowed with a unital associative commutative algebra product, and φ:U⊂Rk→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.

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.

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.

Complex Functions and Residue Theorem

The features and connections of the mappings between complex number fields are thoroughly and methodically summarized and inferred by the complex function.The functional relationship of the mapping between the fields of complex numbers is the subject of study. A complex function uses a complex number as both its independent and dependent variable. The remainder theorem generalizes both the Cauchy's integral theorem and the Cauchy's integral formula. Numerous complex calculations are resolved by the theory of Complex functions and Residue theorem, which has a wide range of applications. This article begins by defining ideas and formulas, demonstrating how formulas connect to concepts, and outlining the procedure and approaches used by mathematicians when working with formulas. The current aspect and the signal aspect are two of the most intimate relationships between Complex Functions and daily living.

Residue Theorem and Its Application

Sometimes it can be frustrating to get the anti-derivatives of some functions. Some definite integrals are difficult, or even impossible, to calculate using the elementary calculus method, especially in the field of complex analysis. In such cases, Cauchy's Residue Theorem might be proper. In this article, we state the difference between Cauchy's Integral Theorem and the Residue Theorem, followed by the definitions of isolated singular point and residue with some examples given after. These are prerequisites for understanding the theorem. A short statement of the content of Cauchy's Residue Theorem has been made, according to which the integral of a function along a positively oriented simple closed contour is related to the residues, which can be obtained through Laurent expansion, of all the limited isolated singular points inside the contour. In addition, using the Cauchy-Goursat theorem, we developed the derivation of CR Theorem. It denotes that the value of a function around a simple closed contour is zero if the function is analytic at all points interior to and on that contour. Then we launch an application and accomplish it in two ways. The first method makes use of the Fourier Matrix, while the second is related to power series. In both of these approaches, we feel the efficacy of Cauchy's Residue Theorem.

Proof of Cauchy’s theorem on Disc

Mathematical analysis known as "complex analysis" examines characteristics of functions, sequences, derivatives of complex numbers, and other objects. When Italian mathematicians Girolamo Cardano and Raphael Bombelli attempted to solve a unique algebra, that is when people first learned about complex numbers. After hundreds of years of growth, complex analysis is now a crucial component of mathematics, physics, and engineering, particularly in the areas of algebraic geometry, fluid dynamics, quantum mechanics, and other related subjects. Complex analysis is a fascinating and esoteric field of study. Mathematicians have created groundbreaking work in this area during the past few centuries. Complex numbers were first defined by mathematicians, who subsequently gradually looked into their algebraic and geometric properties. After hundreds of years, they discovered and studied complex computations. Convergent power series local representations exist for holomorphic functions. This amazing discovery, made by Cauchy between 1830 and 1840, helps to explain the fascinating properties of holomorphic functions. The Cauchy integral theorem is one of the most important concepts in complex analysis. In this paper, a detailed proof of Cauchy’s theorem on a local disc is given.

The Application of Complex Variables Function and Residue Theorem

The history of complex function can be dated back to about 18 century when Euler came up with two equations derived from integrals of complex function. Over the age, mathematicians have been trying to explore and discuss more in this mysterious field, and this new branch gradually became prevalent in the 19 century mathematics like when calculus ruled the 18 century mathematics. In particular, the concept of residue was invented and became an important part in complex function, being applied in some special types of real integral. Also it is a generalization of Cauchy integral theorem and Cauchy integral formula. Nowadays, the area of complex function is filled with crystallization of the wisdom of countless scholars and researchers, and the precious mathematical treasure can also meet the needs of other academic areas like physics and biology. Summarizing the various approaches and examples of application in fields of mathematics and others can be a valuable topic. In this paper, the history of application and some examples for references of complex functions, and introduce the main concepts and formulas about them will be briefly discussed, including the area of complex numbers, analytic function, and residue. There are some relationships between some of the applications to be dig out and considerable real-world problems, and they are supposed to the generalization of those subjects can provide certain enlightenment to people interested in or during the study in fields of complex function.

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.

Bergman theory for the inhomogeneous Cimmino system

We first prove a Cauchy's integral theorem and a Cauchy-type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the mentioned results, dealing in particular with four kinds of weighted Bergman spaces, reproducing kernels, projection and conformal invariant properties.

An analysis of the steepest descent method to efficiently compute the three-dimensional acoustic single-layer operator in the high-frequency regime

Abstract Using the Cauchy integral theorem, we develop the application of the steepest descent method to efficiently compute the three-dimensional acoustic single-layer integral operator for large wave numbers. Explicit formulas for the splitting points are derived in the one-dimensional case to build suitable complex integration paths. The construction of admissible steepest descent paths is next investigated and some of their properties are stated. Based on these theoretical results, we derive the quadrature scheme of the oscillatory integrals first in dimension one and then extend the methodology to three-dimensional planar triangles. Numerical simulations are finally reported to illustrate the accuracy and efficiency of the proposed approach.

Efficient algorithms for integrals with highly oscillatory Hankel kernels

We present an efficient numerical algorithm for approximating integrals with highly oscillatory Hankel kernels. First, using the integral expression of the Hankel function, the integral is transformed into the integral of the trigonometric kernel, and the singularities are separated. According to whether the integration interval contains zeros or not, the integrals are divided into two cases: singular highly oscillatory integrals and non-singular highly oscillatory integrals. For the non-singular case, the Clenshaw–Curtis–Filon rules (CCF) method is used for fast calculation. For the second case, by using the Cauchy integral theorem, the integral is transformed into an infinite integral, which can be calculated by generalized Gaussian–Laguerre rules. The efficiency and accuracy of the new method are verified by numerical examples.

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].)

Analysis of the Propagation in High-Speed Interconnects for MIMICs by Means of the Method of Analytical Preconditioning: A New Highly Efficient Evaluation of the Coefficient Matrix

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

A series representation for Riemann's zeta function and some interesting identities that follow

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $\zeta^{\pm}(s)$ and $\eta^{\pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.

Fuzzy complex numbers: Representations, operations, and their analysis

As a key theoretical basis of fuzzy complex analysis, a new representation of fuzzy complex numbers is proposed in this paper which unifies the exponential form, trigonometric form and algebraic form of fuzzy complex numbers, and some arithmetic operations are investigated. The concepts of strong sum and strong difference are defined in order to simplify the addition and subtraction operations of complex fuzzy numbers and establish the calculus theory of fuzzy complex valued functions. The metric between two fuzzy complex numbers, the derivative, differentiability and analyticity of fuzzy complex-valued functions are defined and characterized. The necessary and sufficient conditions of the analyticity of fuzzy complex-valued functions are investigated. Furthermore, the integral of fuzzy complex-valued functions and Cauchy integral theorem are given and discussed.

Non-monotone waves of a stage-structured SLIRM epidemic model with latent period

We propose and investigate a stage-structured SLIRM epidemic model with latent period in a spatially continuous habitat. We first show the existence of semi-travelling waves that connect the unstable disease-free equilibrium as the wave coordinate goes to − ∞, provided that the basic reproduction number$\mathcal {R}_0 > 1$and$c > c_*$for some positive number$c_*$. We then use a combination of asymptotic estimates, Laplace transform and Cauchy's integral theorem to show the persistence of semi-travelling waves. Based on the persistent property, we construct a Lyapunov functional to prove the convergence of the semi-travelling wave to an endemic (positive) equilibrium as the wave coordinate goes to + ∞. In addition, by the Laplace transform technique, the non-existence of bounded semi-travelling wave is also proved when$\mathcal {R}_0 > 1$and$0 < c < c_*$. This indicates that$c_*$is indeed the minimum wave speed. Finally simulations are given to illustrate the evolution of profiles.

Integration of Heterogeneous Complex Variable Function and Combination Formula

In this paper, we discuss the integration of heterogeneous complex variable function f(zk)=1 (zk-a)n in curve Cj of heterogeneous complex plane Hkby using heterogeneous Cauchy integral theorem, Cauchy integral formula and residue theory. And then obtain two identical equation of combinatorial mathematics by using two different calculation methods.