Laurent Series Expansion Research Articles

A crack in a confocal elliptical inhomogeneity embedded in an infinite matrix subjected to uniform heat flux

We derive a series-form analytical solution to the thermoelastic problem of an insulated and traction-free crack in a confocal elliptical isotropic elastic inhomogeneity embedded in an infinite isotropic elastic matrix subjected to uniform remote heat flux. Using complex variable techniques such as conformal mapping, analytic continuation and Laurent series expansions, the original thermoelastic problem is reduced to an infinite system of linear algebraic equations, the solution of which will yield the mode I and mode II stress intensity factors at the crack tip. An exact closed-form solution is derived when the inhomogeneity and the matrix have identical shear moduli. Detailed numerical results are presented to demonstrate the series-form and closed-form solutions with an emphasis on the particular case of a vanishingly thin inhomogeneity.

Painlevé analysis of the Sasa–Satsuma equation

The Sasa-Satsuma equation is considered. The Painlevé test for partial differential equations with application of the Kruskal variable is used to investigate the necessary condition of integrability of the equation. It is shown that the equation does not pass the Painlevé test in the general case. However, there exist constraints on parameters of the equation, when all Fuchs indices are integers. In these cases the Laurent series expansions of the solution exit and, consequently, there exist analytical solutions of this partial differential equation. We confirm that there are two integrable cases, at which the Sasa–Satsuma equation passes the Painlevé test. We also obtain that there is another set of parameter values at which the equation can be an integrable.

Tropical Laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton polygon associated with the tropical Laurent series. We show that, unlike in the polynomial case, there may be infinitely many tropical roots; moreover, there can be at most two tropical roots of infinite multiplicity. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.

On the boundedness of Euler-Stieltjes constants for the Rankin-Selberg \(L-\)function

Let \(E\) be a Galois extension of \(\mathbb{Q}\) of finite degree and let \(\pi \) and \(\pi'\) be two irreducible automorphic unitary cuspidal representations of \(GL_m(\mathbb{A}_E)\) and \(GL_{m'}(\mathbb{A}_E)\), respectively. Let \(\Lambda(s,\pi\times\widetilde{\pi}')\) be a Rankin-Selberg \(L-\)function attached to the product \(\pi\times\widetilde{\pi}'\), where \(\widetilde{\pi}'\) denotes the contragredient representation of \(\pi'\), and let its finite part (excluding Archimedean factors) be \(L(s,\pi\times\widetilde{\pi}')\). The Euler-Stieltjes constants of the Rankin-Selberg \(L-\)function are the coefficients in the Laurent (Taylor) series expansion around \(s=1+it_0\) of the function \(L(s, \pi \times \widetilde{\pi}')\). In this paper, we derive an upper bound for these constants.

A new method for solving the linearized 1D Vlasov–Poisson system yielding a new class of solutions

We describe a new method for solving the linearized 1D Vlasov–Poisson system by using properties of Cauchy-type integrals. Our method remedies critical flaws of the two standard methods, reveals a previously unrecognized Gaussian-in-time-like decay, and can also account for an externally applied electric field. The Landau approximation involves deforming the Bromwich contour around the poles closest to the real axis due to the analytically continued dielectric function, finding the long-time behavior for a stable system: Landau damping. Jackson's generalization encircles all poles while sending the contour to infinity, assuming its contribution vanishes, which is not true in general. This gives incorrect solutions for physically reasonable configurations and can exhibit pathological behavior, of which we show examples. The van Kampen method expresses the solution for a stable equilibrium as a continuous superposition of waves, resulting in an opaque integral. Case's generalization includes unstable systems and predicts a decaying discrete mode for each growing discrete mode, an apparent contradiction to both the Jackson solution and ours. We show, without imposing additional constraints, that the decaying modes are never present in the time evolution due to an exact cancellation with part of the continuum. Our solution is free of integral expressions, is obtained using algebra and Laurent series expansions, does not rely on analytic continuations, and results in a correct asymptotically convergent form in the case of infinite sums. The analysis used can be readily applied in higher-dimensional, electromagnetic systems and also provides a new technique for evaluating certain inverse Laplace transforms.

Complex variable solution for ground deformation responses considering the grouting pressure of shallow shield

During shield tunneling in shallow strata, ground deformation is highly sensitive to grouting pressure response. Excessive grouting pressure can cause ground uplift, and an excessively low grouting pressure can cause excessive ground settlement, affecting the surrounding environment. Current studies seldom analyze the influence of shield excavation on ground deformation when the grouting pressure is considered. To ensure micro-disturbance of the surrounding environment during shield tunneling, this study presents a complex variable analysis method for analyzing the influence of grouting pressure on surface settlement. This solution maps the semi-infinite strata outside the segment into circular domains by introducing a complex function conformal mapping method. The nonuniform convergence deformation pattern BC-4 was introduced as the boundary condition for the displacement around the tunnel. The Laurent series expansion of the analytical function considering the grouting pressure in the circular domain was substituted into each boundary condition to solve the analytical function. Subsequently, the accuracy of the analytical method was verified by combining it with a finite element numerical method. Finally, the influence of grouting pressure on ground deformation under different working conditions was analyzed through parameterization. This analytical method can provide a theoretical basis for controlling grouting pressure parameters in shield tunnels.

Bounds for Weierstrass Elliptic Function and Jacobi Elliptic Integrals of First and Second Kinds

The Weierstrass elliptic function is presented in connection with the Jacobi elliptic integrals of first and second kinds leading to comparing coefficients appearing in the Laurent series expansion with those of Eisenstein series for the cubic polynomial in the meromorphic Weierstrass function. It is unified in the formulation the Weierstrass elliptic function with Jacobi elliptic integral by considering motion of a unit mass particle in a cubic potential in terms of bounded and unbounded velocities and the time of flight with imaginary part in the complex function playing a major role. Numerical tools box used are the Konrad-Gauss quadrature and Runge-Kutta fourth order method.

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

This study examines the interference problem between screw dislocation and elliptical inclusion in one-dimensional hexagonal piezoelectric quasicrystals. The general solutions are obtained using the complex variable function method and the conformal transformation technique. When the screw dislocation is located outside or inside the elliptical inclusion, the perturbation method and Laurent series expansion are employed to derive explicit analytical expressions for the complex potentials in the elliptical inclusion and the matrix, respectively. Considering four types of far-field force and electric loading conditions, analytical solutions for various specific cases are obtained by using matrix operations. Expressions for the phonon field stress, phason field stress, and electric displacement are given for special cases, including the absence of a dislocation, the presence of an elliptical hole, and the interference between a screw dislocation and circular inclusion, as well as the case of a circular hole. The design and analysis of quasicrystal inclusion structures can benefit from the results of this work.

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

In this paper, the auxiliary equation method is used to study the Weierstrass elliptic function solutions and degenerate solutions of the variable coefficient higher order Schrödinger equation, including Jacobian elliptic function solutions, trigonometric function solutions and hyperbolic function solutions. The types of solutions of the variable coefficient higher-order Schrödinger equation are enriched, and the method of seeking precise and accurate solutions is extended. It is concluded that the types of degenerate solutions are related to the coefficients of the equation itself when the degenerate solutions are obtained from the solutions of the Weierstrass elliptic functions. In addition, the solutions form of the equation is extended from the power series expansion form to the Laurent series expansion form, and the corresponding solutions are obtained. After the conversion formula between the Weierstrass elliptic function solutions and the Jacobian elliptic function solutions is constructed, the Jacobian elliptic function solutions of the higher order Schrödinger equation with variable coefficients are also obtained. These have not been previously studied.

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.

ϵ-expansion of multivariable hypergeometric functions appearing in Feynman integral calculus

We present a new methodology, suitable for implementation on computer, to perform the $\epsilon$-expansion of hypergeometric functions with linear $\epsilon$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $\epsilon$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function. We present illustrative examples of hypergeometric functions in one, two and three variables which are typical of Feynman integral calculus.

Complex number and its discovery history

The operations of complex numbers are the main subject of the mathematical analysis area known as complex analysis. It is also known as the theory of functions of a complex variable. The primary research topic in the field of complex analysis is holomorphic functions. These functions are defined on the complex plane, have differentiable properties, and allow for negative values. The residue theorem, the Cauchy integral formula, the Laurent series expansion, etc. are a few concepts, ideas, and methods that are commonly used in research. In particular, over the years, complex analysis in mathematics, physics, and engineering has been extensively used in algebraic geometry, fluid dynamics, quantum mechanics, and other related areas. Two Italian mathematicians, Girolamo Cardano and Raphael Bombelli, discovered complex numbers in the 16th century while attempting to solve a particular algebra, and Cauchy and Riemann extended it in the 19th century. This essay begin with investigation of arithmetic propertity of comlex numbers and then fully explores history development of complex numbers.

Charaterization of Holomorphic function using power series

Complex analysis is a subfield of mathematical analysis that focuses on the operations of complex numbers. The theory of functions of a complex variable is another name for it. Holomorphic functions, is the key object studied in the filed of complex analysis. These functions can be differentiated, have negative values, and are defined on the complex plane. Some of the ideas, concepts, and techniques frequently utilized in research include the residue theorem, Cauchy integral formula, Laurent series expansion, etc.In particular, algebraic geometry, fluid dynamics, quantum mechanics, and other related sciences have made substantial use of complex analysis in mathematics, physics, and engineering over the years. Girolamo Cardano and Raphael Bombelli, two Italian mathematicians, initially recognized complex numbers while attempting to solve a particular algebra in the 16th century, while Cauchy and Riemann further improved it in the 19th century.This essay explores the arithmetic features of complex numbers firstly. Secondly, a thorough interpretation of the characteristics of holomorphic functions is provided. Finally, using power series methods, the holomorphic function on the disc is characterized.

Analytical Prediction of Tunnel Deformation Beneath an Inclined Plane: Complex Potential Analysis

When excavating a tunnel, the stresses are distributed asymmetrically along the tunnel cross-section. Other factors, particularly slope friction force and excavation speed, can also contribute to the deformation and displacement of a tunnel. Despite this, several authors have used the complex potential method to predict the ground deformation surrounding the tunnel. However, their applicability to the ground response caused by the asymmetric stress distribution around the tunnel wall is analysed in this context. This paper, therefore, proposes an approximate solution on the slope to predict the tunnel cross-section deformation. The solution is based on the complex potential method to predict analytically and numerically the ground deformation around the tunnel. However, two variables called the “complex potential functions” for the Laurent series expansion are used for the stress redistribution to the tunnel boundary conditions. Data from the Qijiazhuang tunnel case are used to justify the proposed analytical solutions. This solution is an essential guide for analyzing deformations in complex geological conditions and structures, such as steeper slopes.

On 𝑞-analogue of Euler–Stieltjes constants

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q q -analogue of the Euler constant and proved the irrationality of certain numbers involving q q -Euler constant. In this paper, we improve their results and prove the linear independence result involving q q -analogue of the Euler constant. Further, we derive the closed-form of a q q -analogue of the k k -th Stieltjes constant γ k ( q ) \gamma _k(q) . These constants are the coefficients in the Laurent series expansion of a q q -analogue of the Riemann zeta function around s = 1 s=1 . Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series ∑ n ≥ 1 σ 1 ( n ) / t n \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1 t > 1 . Finally, we study the transcendence nature of some infinite series involving γ 1 ( 2 ) \gamma _1(2) .

Randomization and the valuation of guaranteed minimum death benefits

In this article, we focus on death-linked contingent claims (GMDBs) paying a random financial return at a random time of death in the general case where financial returns follow a regime switching model with two-sided phase-type jumps. We approximate the distribution of the remaining lifetime by either a series of Erlang distributions or a Laguerre series expansion, whose capability to fit the tail of the observed mortality data turns out to be much better than the commonly used series of exponential distributions.More precisely, we develop a Laurent series expansion of the discounted Laplace transform of the subordinated process at an Erlang distributed time, which leads to explicit formulae for European-type GMDB as well as related risk measures such as the Value-at-Risk (VaR) and the Conditional-Tail-Expectation (CTE). We further concentrate upon path-dependent GMDBs with lookback features like dynamic fund protection or dynamic withdrawal benefits, by relying on a Sylvester equation approach. The advantage of our approaches is that our results are of semi-closed form, avoiding numerical Fourier inversion or Monte-Carlo simulation, leading to fast evaluation. This is necessary in risk-management, in particularly for nested simulation in the framework of Solvency II. Several numerical experiments are included.Our results have implications beyond life-insurance and GMDBs, namely in all situations where randomization or Erlangization replaces known quantities, like, for example, model parameters, by random variables. In Finance, it is for example well-known that a random maturity time leads to much more convenient valuation formulas that well approximate its non-random counterpart.

EXPANDING THE LAURENT SERIES WITH ITS APPLICATIONS

In Nepal, there are many mathematics subjects taught at university level. Among them, complex analysis is the most powerful. In complex analysis, the Laurent series expansion is a well-known subject because it may be used to find the residues of complex functions around their singularities. It turns out that computing the Laurent series of a function around its singularities is an effective way to calculate the integral of the function along any closed contour around the singularities as well as the residue of the function. Learning the Laurent series concepts can be difficult, and many students struggle to develop adequate understanding, reasoning, and problem-solving skills. Therefore, this article presents multiple practical examples where the Laurent series of a function is found and then utilized to compute the integral of the function over any closed contour around the singularities of the function, based on the theory of the Laurent series.

Host movement, transmission hot spots, and vector-borne disease dynamics on spatial networks.

We examine how spatial heterogeneity combines with mobility network structure to influence vector-borne disease dynamics. Specifically, we consider a Ross-Macdonald-type disease model on n spatial locations that are coupled by host movement on a strongly connected, weighted, directed graph. We derive a closed form approximation to the domain reproduction number using a Laurent series expansion, and use this approximation to compute sensitivities of the basic reproduction number to model parameters. To illustrate how these results can be used to help inform mitigation strategies, as a case study we apply these results to malaria dynamics in Namibia, using published cell phone data and estimates for local disease transmission. Our analytical results are particularly useful for understanding drivers of transmission when mobility sinks and transmission hot spots do not coincide.

A fixed-point current injection power flow for electric distribution systems using Laurent series

This paper proposes a new power flow (PF) formulation for electrical distribution systems using the current injection method and applying the Laurent series expansion. Two solution algorithms are proposed: a Newton-like iterative procedure and a fixed-point iteration based on the successive approximation method (SAM). The convergence analysis of the SAM is proven via the Banach fixed-point theorem, ensuring numerical stability, the uniqueness of the solution, and independence on the initializing point. Numerical results are obtained for both proposed algorithms and compared to well-known PF formulations considering their rate of convergence, computational time, and numerical stability. Tests are performed for different branch R/X ratios, loading conditions, and initialization points in balanced and unbalanced networks with radial and weakly-meshed topologies. Results show that the SAM is computationally more efficient than the compared PFs, being more than ten times faster than the backward–forward sweep algorithm.

A series representation of Euler–Stieltjes constants and an identity of Ramanujan

We derive a series representation of the generalized Stieltjes constants which arise in the Laurent series expansion of partial zeta function at the point s=1. In the process, we introduce a generalized gamma function and deduce its properties such as functional equation, Weierstrass product and reflection formulas along the lines of the study of a generalized gamma function introduced by Dilcher in 1994. These properties are used to obtain a series representation for the k-th derivative of Dirichlet series with periodic coefficients at the point s=1. Another application involves evaluation of a class of infinite products of which a special case is an identity of Ramanujan.