Cauchy Residue Research Articles

Wave resonances and the time-dependent capillary gravity wave motion

The interconnection of time and frequency domains for capillary gravity wave motion in the presence of current is discussed in this article. The general time-dependent problem is solved using Green’s function technique, and the asymptotic solution is derived using the method of stationary phase for large time and space. Also, the frequency domain solution is derived as a special case using the Cauchy Residue theorem. Different types of wave resonances like Trapping, Blocking and Bragg resonances are discussed. The existence of the trapped mode below the cutoff frequency is justified theoretically, and numerical results are obtained using the multipole expansion method. The blocking and Bragg resonances are analyzed above the cutoff frequency. It is found that in the presence of current, when the ripple wavenumber of the bottom undulation equals twice the cosine angle of incidence of wave times the wavenumber of the wave, Bragg resonance occurs. It is found that three propagating modes exist in the case of wave blocking, and the trapped modes exist only for the first propagating mode. Furthermore, because of the negative group velocity inside the blocking zone, the Bragg reflection increases while decreasing outside. The effect of current on the wave energy propagation in the form of group velocity is analyzed and the same is verified in the case of time-dependent problem.

On integral theorems and their statistical properties

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The integral theorems provide natural estimators of density functions via Monte Carlo methods. Assessment of the quality of the density estimators can be used to obtain optimal cyclic functions, alternatives to the sin function, which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis.

Applications of Residue Theorem to Some Selected Integrals

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases.

Conversion from Shear Creep Compliance to Relaxation Modulus of Asphalt Binder Using Cauchy Residue Method and Experimental Validation

Conversion from Shear Creep Compliance to Relaxation Modulus of Asphalt Binder Using Cauchy Residue Method and Experimental Validation

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.

Cauchy Residue Theorem and K-residue Theorem

Due to negative numbers do not have square roots, scientists introduced complex numbers, which are more abstract compared with real numbers. Residue Theorem has a very significant status in complex analysis – it can be used to simplify difficult integrals. In this article, Cauchy’s Residue Theorem is first introduced with definition and proof. Then the author shows two examples of using the theorem to solve problems, which can help readers to better explain the theorem. After that, K-residue Theorem is introduced, which is a special kind of Residue Theorem, with a special function called K-function. Finally, the theorem is used to solve a problem. These problems are difficult to solve without Residue Theorem. Through the analysis, it is easy to find the importance of Residue Theorem: it is very useful, not only in math but also in many other fields like physics, biology, and chemistry. Moreover, it still has more potential for people to develop.

Cauchy Theorem and Cauchy Residue Theorem

Cauchy Theorem and Cauchy Residue Theorem

A simple, flexible technique for RF cavity wake-field calculations

It is typical in the accelerator field to model machine components, especially RF cavities, as parallel RLC resonators. To properly model wake-fields, knowledge of the time-domain voltage resulting from beam excitation is often necessary. While analytical and quasi-analytical expressions are available to accomplish this for common bunch distributions such as the Gaussian, analogous results for less standard distributions can be difficult or computationally-taxing to obtain using direct methods, which opens the door for the development of a more generalized technique. In this paper, a formulation is created that allows for the simple computation of the time-domain voltage waveform of an RLC resonator. The formulation uses the Cauchy Residue Theorem to extract the convolution result from the Fourier Domain, and if current distribution Fourier Transform has no poles, knowledge of its value is only required at one specific evaluation point. This greatly simplifies the computation of the time domain voltage for a large amount of bunch distributions both common and uncommon. Accuracy considerations for this technique and the approximation of accelerator components as RLC resonators are also discussed, resulting the development of a figure of merit for quantifying the robustness of this type of approximation.

Application of generalised functions and Fourier’s integral transformation in modelling the effect of a moving load on a beam resting on an elastic foundation

The paper is dedicated to modelling the behaviour of an infinitely long beam resting on a solid elastic foundation under the concentrated moving load effect. The modelling is carried out in the context of determining the high-speed vehicle stock impact on the track structure. When developing the method of problem solving, the known elasticity theory factors, wave-propagation theory, complex variable theory, as well as integral transformations of generalised functions were used. The mobile concentrated force is represented by the Dirac delta function. The Fourier’s integral transformation is used to solve the equation of motion and determine the beam defect. Contour integration using Cauchy residue theorem is applied to calculate the defect values in the original space. The general dependence form the beam defect on the load velocity is determined. It is shown that the defect values in the beam increase with moving load increasing speed. It is shown how the moving load velocity affects the character of the beam defect time dependence. Solutions are given for Euler-Bernoulli, Rayleigh and Timoshenko models of beams on an elastic base. The considered mathematical models behaviour is compared. It is shown that the character of beam interaction with the moving load, the defect values, as well as the reaching critical velocities character do not differ significantly for the considered beam models. The character of reaching the maximum defect in the application area of the mobile load is analysed. It is shown that the maximum defect values of beams on an elastic base occur after the moving load. The presented methodology can be used to evaluate the impact of rolling stock on the track structure, including the design of high-speed lines.

Analytical solutions of linear delay-differential equations with Dirac delta function inputs using the Laplace transform

In this paper, we propose a methodology for computing the analytic solutions of linear retarded delay-differential equations and neutral delay-differential equations that include Dirac delta function inputs. In numerous applications, the delta function serves as a convenient and effective surrogate for modeling high voltages, sudden shocks, large forces, impulse vaccinations, etc., applied over a short period of time. The solutions are obtained using the Laplace transform method, in conjunction with the Cauchy residue theorem. The accuracy of these solutions are assessed by comparing them with the ones provided by the method of steps. Numerical examples illustrating the methodology are presented and discussed. These examples show that the Laplace transform solution is very reliable for linear retarded delay-differential equations, because the analytic solution, for a single delta function input, is continuous. However, for linear neutral delay-differential equations with a delta function input the analytic solution is discontinuous. Consequently, the well-known Gibbs phenomenon is observed in the vicinity of the discontinuities. However, for neutral delay differential equations, we show that in some cases, the magnitude of the jumps at the discontinuities decrease, as time increases. Therefore, the Gibbs phenomenon of the Laplace solution dissipates.

Analytic solutions of linear neutral and non-neutral delay differential equations using the Laplace transform method: featuring higher order poles and resonance

In this article, we extend the Laplace transform method to obtain analytic solutions for linear RDDEs and NDDEs which have real and complex poles of higher order. Furthermore, we present first-order linear DDEs that feature resonance phenomena. The procedure is similar to the one where all of the poles are order one, but requires one to use the appropriate modifications when using Cauchy’s residue theorem for the poles of higher order. The process for obtaining the solution relies on computing the relevant infinite sequence of poles and then determining the Laplace inverse, via the Cauchy residue theorem. For RDDEs, the poles can be obtained in terms of the Lambert W function, but for NDDEs,the complex poles, in most cases, must be computed numerically. We found that an important feature of first-order linear RDDES and NDDES with poles of higher order is that it is possible to incite the resonance phenomena, which in the counterpart ordinary differential equation cannot occur. We show that despite the presence of higher order poles or resonance phenomena, the solutions generated by the Laplace transform method for linear RDDEs and NDDEs that have higher order poles are still accurate.

Development of a new dynamic hysteresis model for magnetorheological dampers with annular-radial gaps considering fluid inertia and compressibility

Modeling the inherent non-linear dynamic hysteresis behavior of magnetorheological (MR) fluid dampers is a fundamental necessity for their practical implementation. Quasi-static models, have been developed under very low frequency and small amplitude excitation conditions, are only beneficial in the early stages of MR dampers design and are sufficient for manufacturing requirements. These models, however, have failed to predict the non-linear hysteresis behavior of MR dampers (i.e., force-displacement and force-velocity characteristics) under low to moderate ranges of frequency and displacement amplitudes. The main contribution of the present work is to formulate a novel physic-based dynamic model to predict the non-linear behavior of MR dampers featuring an annular-radial bypass valve under different applied currents, and low to moderate input excitations. The proposed model incorporates unsteady fluid behavior and the fluid compressibility effect on the hysteretic response without relying on experimental data to evaluate the model parameters. To derive the model, the velocity profile and the pressure drop have been derived by solving the fluid momentum equation via the Laplace transform technique and Cauchy residue theory. The continuity equation has been employed for considering the compressibility contribution to hysteretic response based on the variation of bulk moduli. The predicted results for the force-displacement and force-velocity behavior of the MR damper were subsequently compared with the experimental results. Results suggest the effectiveness of the proposed model for predicting the dynamic hysteresis characteristics of the annular-radial MR fluid dampers under the relatively sufficient variation of mechanical and magnetic loading conditions considered.

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.

Cauchy Residue Theorem’s Application in Improper integrals

Definite integrals are an essential tool for understanding and calculating many aspects of the natural world. An improper integral, one type of definite integral, has either an infinite interval or an integrand that is not defined at one or more points within the interval of integration. In this research, improper integrals are the main concepts that will be discussed. The function can be expressed in terms of its Laurent series expansion–a general form or representation for analytic functions that includes both negative and positive power series of (z – singularity)–about each of its isolated singularities within the contour. And then, finding the singularities that are inside the contour based on the contour function. Calculating the residue of the singularities. Then, substituting the residue with the calculated value, adding all the residue together. Lastly, the result multiplies by, which is the result, namely the integral of the original functions. With the help of the other two methods, Keyhole Contour, and principal values, it is possible to evaluate the integrals and determine the domain of the function. Keyhole Contour separates the function’s domain and evaluates real integrals. And principal values can be used for determining the specific range of the function, so a single value can be chosen.

Construction of Fourier Series Expansion of Apostol-Frobenius-Type Tangent and Genocchi Polynomials

In this study, the Fourier series expansions of the Apostol-Frobenius type of Tangent and Genocchi polynomials of higher order are derived using the Cauchy residue theorem. Some novel and intriguing results are obtained by applying the Fourier series expansion of these types of polynomials.

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 Residue Theorem

Cauchy's Residue Theorem, commonly known as the residue Theorem, is a key theorem in complex variables because it enables people to determine the enclosed curve line of integrals for functionalities. Real integrals and infinite series may also be calculated using it. Our ability to handle complicated analysis is improved by the residue theorem. By connecting the internal Ehrhart polynomials to the closures, we illustrate these tetrahedron Ehrhart-Macdonald reciprocity laws. We calculate the Earhart coefficient of the codimensions to demonstrate our methodology. We conclude by demonstrating how to use our ideas to locate any paste applied with a convex lattice.

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.

Application and Analysis of The Residue Theorem

This paper studies the derivation of the residue theorem and its application. The residue theorem is an important part of the theory of complex functions and plays a critical role in the development and application of complex functions. At the same time, the residue theorem advances the method of solving the value of definite integrals to a new stage. The issue of the Cauchy residue theorem has received considerable attention. Also, residue theorem lays the foundation for the development of integral theory, and it has been widely used in many scientific areas. The goal of this study is to investigate the properties and applications of the residue theorem. To this end, the strategies to deduce the residue theorem are summarized. Then this paper uses several examples that relate to the residue theorem to help people understand the residue theorem more deeply. The integrals under consideration include trigonometric integrals and integrals involving power functions.