A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

Continued fractions and q-series generating functions for the generalized sum-of-divisors functions

A bstract In this paper, we present analytical results for the two-loop QCD corrections to the production of two partons or a photon and a parton in hadronic collisions, mediated by loops of massive quarks. These amplitudes involve Feynman integrals defined on an elliptic curve. We compute them by generalizing our recent results for the production of two photons to include additional crossings of the corresponding master integrals, which we compute in terms of the same basis of independent iterated integrals. We discuss the analytical properties of the amplitudes, highlighting the cancellations of a large number of elliptic differential forms in their finite remainders. Finally, we elaborate on a strategy for their numerical evaluation based on generalized series expansions at singular points of the physical amplitude, through the introduction of suitable sets of variables that allow us to resolve all singularities.

A detailed consideration is given of a recent reformulation of Hund's rules in terms of inequalities. The interpretation of Hund's rules is reconsidered assuming that each energy level is known exactly in the restricted Hartree-Fock approximation. Some general theorems and series expansions are derived for energy differences between atomic states. As is now well known, the state of lower energy in a multiplet has the higher expectation value for the electron-electron repulsion, especially in neutral systems. The behaviour of this repulsion is analysed in terms of the Fermi hole and the contraction of the wavefunction which one finds for the state of lowest energy. Numerical examples are given for the 1P and 3P states of the He isoelectronic series, the 1S, 1D and 3P states of the C and O series and the 2P, 2D and 4S states of the N series.

In this chapter we move away from the Fourier series and the general orthogonal series expansions. We will consider the appearance of the Gibbs phenomenon in the piecewise-linear approximation of functions with jump discontinuities that was first considered by Foster and Richards [28]. In this case, they demonstrated that the size of the overshoot is larger, i.e. about 13.4% compared to the classical 8.95% value of the Fourier trigonometric series and integrals approximations. For both cases here the usual measure of L 2 (or the least square)-approximation is employed. They pointed out that the (lower) overshoot of 8.95% is not exclusive for the trigonometric polynomial, but rather due to the latter being the best least square approximation. This piecewise-linear approximation is seen as a special case of using splines of degree one approximation. Richards [29] extended this analysis to high order splines where he showed that the size of the overshoot decreases with increasing the order m of the spline ∅ m+i (x) considered. Indeed it turns out that the size of the overshoot approaches the classical one of 8.95% as the order m + 1 (degree m) of the employed spline approaches infinity. This result was sought and proved almost completely by Richards [29], then was finalized later by Shim and Volkmer [30] and Foster and Richards [31].

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

<abstract><p>In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.</p></abstract>

Piezoelectric materials have been used widely in applications for sensing and actuation purposes in recent years. As piezoelectric sensors and actuators usually are thin in comparison to the relevant wavelengths, the analysis of thin piezoelectric layers has attracted considerable research interest. The present work considers the derivation of new plate equations and the corresponding edge boundary conditions for an orthotropic piezoelectric plate. Here, power series expansions in the thickness coordinate for the displacements and the electric potential are stated, that lead to recursion relations among the expansion functions. These recursion relations are important as all fields hereby can be expressed in a finite number of expansions functions without performing any truncations. Moreover, the recursion formulas involve no approximations since they stem from the definition of the series expansions and are as such exact. Using the recursion relations in the boundary conditions on the surfaces and along the edges, a set of plate equations and the corresponding edge boundary conditions are obtained in a systematic fashion. Hence the surface boundary conditions constitute the plate equations of motion, and as such the lateral boundary conditions are exactly fulfilled. These plate equations can be truncated to any order in the thickness and it is believed that they are asymptotically correct, based on experience for other structures. Contrary to most traditional theories, there is no need to assume kinematical simplifications based on various engineering assumptions; neither for the equations of motion nor for the corresponding edge boundary conditions. Note that various sorts of series expansion procedure have been developed by others, but there are several differences in the derivation procedure among these works, such as in the series expansion formulation, the development of recursion relations or in the truncation process. The plate theory developed is given explicitly for flexural motion using a low order truncation. These equations of motion resemble the Mindlin type plate equations, albeit the present theory involves some further terms as well as no correction factors. The accuracy of the plate theory is displayed for the displacement, electric and stress fields, where comparisons are made with exact and traditional approximate theories. These results show that the present asymptotic plate theory renders considerably more accurate results than the Kirchhoff and the Mindlin theories.

The main objective of this paper is to describe a class of functional series expansions, known as Fliess operators, which admit inputs from a ball in an L p space as well as Poisson random processes. Conditions are given under which these functional series expansions converge absolutely in the mean. Then, it is shown that a continuous-time switched input-affine nonlinear system with a Poisson switching signal can be represented as a Fliess operator and that for certain cases a closed form solution can be obtained in terms of Poisson integrals.

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and for Gaussian hypergeometric functions F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) , in particular, if the parameters are also viewed as variables.

In this paper, noise estimation based on series expansion of orthogonal functions is proposed. The proposed method searches speechless frequency regions and estimates the noise spectrum from the searched speechless frequency regions. The proposed method can adapt to changing of noise power without estimation delay. Experimental results show that the proposed method provides a good performance against time-varying noise.

A Taylor power series is a powerful mathematical tool, which can be used to express an inverse function especially if it is given in an implicit form. This is for example the case for the inverse Langevin function, which is an indispensable ingredient of full-network rubber models. In the present paper, we propose a simple recurrence procedure for calculating Taylor series coefficients of the inverse function. This procedure is based on the Taylor series expansion of the original function and results in a simple recurrence formula. This formula is further applied to the inverse Langevin function. The convergence radius of the resulting series is evaluated. Within this convergence radius the obtained approximation of the inverse Langevin function demonstrates better agreement with the exact solution in comparison to different Padé approximants.

In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants from cancellation of odd- and even-length walk terms. We report some progress, but not a complete resolution, of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. Furthermore, we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.

Power series expansions for spheroidal wave functions with small arguments

Series expansion of the period function and representations of Hecke operators

In this paper, a method is proposed for improving the accuracy of the stability criterion by enlarging the estimation of the stability region determined by means of Lyapunov's direct method using series expansion, and by putting it closer to the actual stability region. The Lyapunov function obtained by energy integration is used as the first estimation. The method is applied to a single-machine infinite bus power system, and the stability domain in the δ-ω plane is shown together with the actual stability domain obtained by numerical integration. It is shown that the application of the proposed method results in a considerable improvement in the stability boundary estimation over that given by the original Lyapunov function. Zubov's method, which obtains an accurate stability boundary using the series expansion, is related to the method proposed in this paper. The last section is, therefore, devoted to comparing the two methods and examining the possibility of their application to multi-machine power systems.