Integral Theorem Research Articles

The Application of Residue Theorem in The Calculations of Real Integrals

Complex analysis focuses on the functions of complex variables. The research on complex analysis began in 19th century by mainly Cauchy, Riemann, and Weierstrass. In 1825, Cauchy established Cauchy’s integral theorem, indicating that the value of a complex function at a point within a closed contour is entirely dependent on values of the functions at points on the contour . Cauchy’s integral theorem is a significant theorem in complex analysis, which was then further developed into Cauchy’s residue theorem. In this paper, the authors introduce the fundamentals of complex analysis. In Introduction, the authors discuss the history of complex analysis and residue theorem and summarizes the content of published articles and books. In Method, the authors explain the definitions and fundamental properties of complex functions. In Result, the authors use examples of different forms of real integrals to demonstrate the application of residue theorem in the calculation of real integrals.

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.

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.

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.

Convolved action principles for couple stress elastodynamics

A novel set of variational principles and integral theorems is developed for consistent couple stress elastodynamics, including the principle of virtual convolved action, the principle of stationary convolved action, the theorem of dynamic reciprocity and the integral representation theorem. Both time-domain and Laplace-domain versions are presented. Most notably, all of these are based on the concept of convolved action, involving temporal convolutions, and are applicable to the most general case of bi-anisotropic couple stress materials. Furthermore, the principle of virtual convolved action extends beyond elastodynamics to viscoelastic and plastic material response. In addition, the two-dimensional infinite space fundamental solutions and kernel functions are derived in the Laplace-domain and used to formulate a new boundary element method (BEM) for transient response, which also originates from a convolved action. The resulting BEM is applied to four prototype examples to demonstrate the validity and robustness of the method, and to explore interesting behavior of consistent couple stress elastodynamics in the isotropic case. Although the focus here is solely on couple stress elastodynamics, the convolved action framework extends to a broad range of time-dependent problems, including those of a multiphysical nature.

Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner (with an appendix by Emilio Corso)

We apply a method inspired by Ratner’s work on quantitative mixing for the geodesic flow [29] and developed by Burger [11] to study ergodic integrals for horocycle flows. We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni [15], and we show that the coefficients in the asymptotic expansion are Hölder continuous with respect to the base point. Furthermore, we provide short and streamlined proofs of the spatial limit theorems of Bufetov and Forni [10] and, in an appendix by Emilio Corso, of a temporal limit theorem by Dolgopyat and Sarig [12].

On calculation of the exact Lyapunov exponent of affine Boole transformations

We show an elementary way to calculate the exact Lyapunov exponent of affine Boole transformations using the interchangeability theorem for differentiation and integration due to Leibniz.

A Fubini type theorem for rough integration

Jointly controlled paths as used in Hairer and Gerasimovičs (2019), are a class of two-parameter paths Y controlled by a p -rough path X for 2 \leq p < 3 in each time variable, and serve as a class of paths twice integrable with respect to X . We extend the notion of jointly controlled paths to two-parameter paths Y controlled by p -rough and \tilde{p} -rough paths X and \tilde{X} (on finite dimensional spaces) for arbitrary p and \tilde{p} , and develop the corresponding integration theory for this class of paths. In particular, we show that for paths Y jointly controlled by X and \tilde{X} , they are integrable with respect to X and \tilde{X} , and moreover we prove a rough Fubini type theorem for the double rough integrals of Y via the construction of a third integral analogous to the integral against the product measure in the classical Fubini theorem. Additionally, we also prove a stability result for the double integrals of jointly controlled paths, and show that signature kernels, which have seen increasing use in data science applications, are jointly controlled paths.

Discrete exterior calculus for meshes with concyclic polygons

Discrete exterior calculus (DEC) is a numerical method for solving partial differential equations on meshes with applications in computer graphics, numerics, and physical simulations. It discretizes PDEs in a way such that important integral theorems hold exactly instead of being approximated. The drawback of the method is that it has strong requirements on the mesh. While the narrow range of admissible meshes mentioned in the original work could be widened, current methods still exclude an essential category of meshes, i.e., meshes with concyclic triangle pairs. Such meshes are common, as many synthetic meshes, e.g., triangulations of CAD models, handcrafted 3D models, and even results of surface meshing algorithms, contain such triangle pairs. Our paper describes an approach that allows us to use meshes with arbitrary triangulations that may contain concyclic triangle pairs by defining DEC operators for concyclic polygons.

Quantum local-equilibrium approach to dissipative hydrodynamics.

The macroscopic hydrodynamic equationsare derived for many-body systems in the local-equilibrium approach, using the Schrödinger picture of quantum mechanics. In this approach, statistical operators are defined in terms of microscopic densities associated with the fundamentally conserved quantities and other slow modes possibly emerging from continuous symmetry breaking, as well as macrofields conjugated to these densities. Functional identities can be deduced, allowing us to identify the reversible and dissipative parts of the mean current densities, to obtain general equationsfor the time evolution of the conjugate macrofields, and to establish the relationship to projection-operator methods. The entropy production is shown to be nonnegative by applying the Peierls-Bogoliubov inequality to a quantum integral fluctuation theorem. Using the expansion in the gradients of the conjugate macrofields, the transport coefficients are given by Green-Kubo formulas and the entropy production rate can be expressed in terms of quantum Einstein-Helfand formulas, implying its nonnegativity in agreement with the second law of thermodynamics. The results apply to multicomponent fluids and can be extended to condensed matter phases with broken continuous symmetries.

The Jacobi - Poisson Method of Building First Integrals for System of Ordinary Differential Equations

In this paper, the Jacobi-Poisson method of building first integrals by known integral manifolds for Hamiltonian differential systems is developed. The existence theorem of first integrals in the form of Poisson brackets from known integral manifolds for Hamiltonian systems is proved. The statement of finding additional first integrals by known first integrals and integral manifolds for Hamiltonian systems is given. In the case of polynomial Hamiltonian systems, we use the concept of partial integrals and concretize ours results. The Jacobi - Poisson method of construction first integrals in the form of Poisson brackets from known integral characteristics (integral manifolds, partial integrals, and first integrals) for general ordinary differential systems is proposed. Statements of finding first integrals for an ordinary differential system by integral characteristics of the auxiliary Hamiltonian system are obtained. The generalized Poisson theorem of construction first integrals is indicated. And, the question of the existence of partial integrals and first integrals in the form of Poisson brackets for ordinary polynomial differential systems is investigated. The results obtained in this paper can be used in the analytical theory of differential equations and in the analytical mechanics.

Maxwell’s equations revisited – mental imagery and mathematical symbols

Using Maxwell’s mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations $\operatorname{curl} \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, $\operatorname{curl} \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{j}$, $\operatorname{div} \mathbf{D} = \varrho $, $\operatorname{div} \mathbf{B} = 0$, which together with the constituting relations $\mathbf{D} = \varepsilon _0 \mathbf{E}$, $\mathbf{B} = \mu _0 \mathbf{H}$, form what we call today Maxwell’s equations. Main tools are the divergence, curl and gradient integration theorems and a version of Poincare’s lemma formulated in vector calculus notation. Remarks on the history of the development of electrodynamic theory, quotations and references to original and secondary literature complement the paper.

An Alternative Numerical Scheme to Approximate the Early Exercise Boundary of American Options

This paper deals with a new numerical method for the approximation of the early exercise boundary in the American option pricing problem. In more detail, using the mean-value theorem for integrals, we provide a flexible algorithm that allows for reaching a more accurate numerical solution with fewer calculations rather than other previously described methods.

Stokes, Gibbs, and Volume Computation of Semi-Algebraic Sets

We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain infinite-dimensional linear program (LP). At each step one solves a semidefinite relaxation of the LP which involves pseudo-moments up to a certain degree. Its dual computes a polynomial of same degree which approximates from above the discontinuous indicator function of the set, hence with a typical Gibbs phenomenon which results in a slow convergence of the associated numerical scheme. Drastic improvements have been observed by introducing in the initial LP additional linear moment constraints obtained from a certain application of Stokes’ theorem for integration on the set. However and so far there was no rationale to explain this behavior. We provide a refined version of this extended LP formulation. When the set is the smooth super-level set of a single polynomial, we show that the dual of this refined LP has an optimal solution which is a continuous function. Therefore in this dual one now approximates a continuous function by a polynomial, hence with no Gibbs phenomenon, which explains and improves the already observed drastic acceleration of the convergence of the hierarchy. Interestingly, the technique of proof involves recent results on Poisson’s partial differential equation (PDE).

Analytic Fourier–Feynman transforms associated with bounded linear operators on abstract Wiener spaces

In this paper, we introduce the concepts of the analytic Feynman integral and the analytic Fourier–Feynman transform associated with bounded linear operators on abstract Wiener spaces. We then investigate Fubini theorems for the analytic Feynman integrals and the transforms. The Fubini theorems for the transforms investigated in this paper are to express the iterated Fourier–Feynman transform associated with bounded linear operators as a single Fourier–Feynman transform.

Mathematical study on the potential flow past a vertical submerged flexible plate of non-uniform thickness

The present work emphasizes on the mathematical study of two-dimensional potential fluid flow past a vertical flexible plate of non-uniform thickness submerged in a water domain of finite depth. The boundary value problem associated with the described model is solved by two mathematical techniques while considering Kirchhoff’s thin plate theory. The first method is based on a perturbation analysis involving a small parameter ϵ, while the second method directly employs the classical method of integration and Green’s integral theorem to the governing equations to obtain the solution. Both the techniques result into two different systems of integral equations which are solved numerically using appropriate approximations. It has been established numerically that for smaller thickness variations depicted by particular values of ϵ (ϵ≤0.001), the two methods yield the same numerical results for all the hydrodynamic quantities. The correctness of the present results is verified by comparing them graphically with the ones already existing in the literature. Effects of the non-uniform plate thickness on various hydrodynamic parameters are presented graphically by choosing various thickness profiles. Also, comparisons are made with the results of constant thickness elastic plate to distinguish its hydrodynamic behaviour from the one having non-uniform thickness.

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.

Driven particle in a one-dimensional periodic potential with feedback control: Efficiency and power optimization.

A Brownian particle moving in a staircaselike potential with feedback control offers a way to implement Maxwell's demon. An experimental demonstration of such a system using sinusoidal periodic potential carried out by Toyabe etal. [Nat. Phys. 6, 988 (2010)1745-247310.1038/nphys1821] has shown that information about the particle's position can be converted to useful work. In this paper, we carry out a numerical study of a similar system using Brownian dynamics simulation. A Brownian particle moving in a periodic potential under the action of a constant driving force is made to move against the drive by measuring the position of the particle and effecting feedback control by altering potential. The work is extracted during the potential change and from the movement of the particle against the external drive. These work extractions come at the cost of information gathered during the measurement. Efficiency and work extracted per cycle of this information engine are optimized by varying control parameters as well as feedback protocols. Both these quantities are found to crucially depend on the amplitude of the periodic potential as well as the width of the region over which the particle is searched for during the measurement phase. For the case when potential flip (i.e., changing the phase of the potential by 180^{∘}) is used as the feedback mechanism, we argue that the square potential offers a more efficient information-to-work conversion. The control over the numerical parameters and averaging over large number of trial runs allow one to study the nonequilibrium work relations with feedback for this process with precision. It is seen that the generalized integral fluctuation theorem for error-free measurements holds to within the accuracy of the simulation.