Series Expansion Of Functions Research Articles

Rational approximation to harmonic functions

Pade-type approximation is the rational function analogue of Taylor’s polynomial approximation to a power series. A general method for obtaining Pade-type approximants to Fourier series expansions of harmonic functions is defined. This method is based on the Newton-Cotes and Gauss quadrature formulas. Several concrete examples are given and the convergence behavior of a sequence of such approximants is studied.

A full-wave modal analysis of arbitrarily shaped waveguide discontinuities using the finite plane-wave series expansion

A full-wave electromagnetic model for analyzing waveguide discontinuities of arbitrarily shaped piecewise planar boundaries is presented. The analysis is facilitated by using the finite plane-wave series expansion of circular cylindrical modal functions. Since electromagnetic fields on each of the planar boundary surfaces of the inhomogeneous region are expressed in terms of plane-wave modal functions, the complete solution is carried out analytically without any numerical integration. To verify the formulation, a number of practical waveguide components are analyzed. The calculated results are compared with other full-wave electromagnetic models. Excellent agreement is obtained for all the cases.

On the analytical summation of Fourier series and its relation to the asymptotic behaviour of Fourier transforms

Forms of the Poisson summation formula (PSF) appropriate for the summation of semi-infinite and infinite Fourier series are derived. Application of these results to the acceleration of convergence of various types of series with monotonically decreasing coefficient functions yields transformed series with terms that decay either exponentially or with the inverse first or second power of the index variable. These two very different convergence properties are explained in terms of the asymptotic properties of the relevant Fourier transforms, which are in turn related to the power series expansions of the summand functions in the original Fourier series. The result is that the Poisson summation formula works best for Fourier cosine series in which the summand functions are expansible in even powers, and for Fourier sine series in which the summand functions have power series with odd powers. Here, application of the PSF produces series of terms that decay exponentially with increasing argument x. In contrast, application of the semi-infinite version of the PSF to Fourier cosine series of terms with odd-power expansions, or to Fourier sine series of terms with even-power expansions yields transformed series involving functions of the form , which decay approximately as . If the summand function in the Fourier series has a power series with both even and odd powers, the transformed series involves sine and cosine integral functions, which decay approximately as . Fourier series of these last three types in general require additional acceleration, for example, by application of the Kummer transformation.

Detail Study of the Resonant Transition in a Paul Trap

The resonant transition in the Paul trap is studied by the time-space transformation and function series expansion. A new selection rule for the transition in the trap is derived strictly. The probability in the ground state and some transition rates to the excited states are calculated. The comparison with the former studies shows that our results are more accurate and credible.

Sound absorption by an orifice plate in a flow duct

In this paper the acoustic absorption due to an orifice plate in a duct supporting a mean flow is studied theoretically. Absorption takes place as the acoustic field energizes a vortex field which is generated at the orifice rim. A linearized approximation is made to the absorption mechanism. This work presents an analytical extension of Howe [Proc. R. Soc. London, Ser. A 366, 205–233 (1979)]. The latter deals with unsteady high Reynolds number flow through a circular aperture in a thin infinite screen. To the author’s knowledge, such an extension has not been previously made. The problem is formulated analytically insofar as it is possible and numerical results are presented. A Green’s function series expansion is used in the formulation. A difficulty arises with the convergence of this expansion. It is solved by a renormalization technique, which has been developed for this problem. The technique appears to be a novel method for dealing with convergence problems associated with term by term differentiation of Green’s function series expansions. To provide a check on the solution, it is shown that when the radius of the duct tends to infinity the present expression for the Rayleigh conductivity of the orifice plate limits to the expression obtained by Howe for an aperture in a thin infinite screen. With respect to the numerical results, it appears that for orifice mean flow Mach numbers ≲0.2, an orifice to duct open area ratio of 0.3 provides near optimal average absorption, for the band of frequencies limited by the first symmetric mode cutoff frequency.

Deformations of an elastic half plane with a circular cavity

An analytical solution is given of the class of problems of an elastic half plane with a circular cavity, loaded on the cavity boundary. The solution uses complex variables, with a conformal mapping onto a circular ring. The coefficients in the Laurent series expansions of the stress functions can be expressed into a single constant by a system of recurrent relations, obtained from the boundary conditions. The remaining constant can be determined from the requirement of convergence of the series. For the case of a uniform radial stress at the cavity boundary the solution can be given in closed form, confirming known results for the stresses, but also giving simple explicit expressions for the displacements.

A Study of the Characteristics of the Wave Packets of a Paul-Trapped Ion

Based on the former studies, the time-dependent characteristics of the wave packets of a Paul-trapped ion are investigated explicitly in cylindrical coordinates by the method of function series expansion (FSE). It is shown that the present work relates the quantum motion of a Paul-trapped ion to the orbit of the classical motion and presents a more clear physical picture than the former studies.

High-temperature series expansion of the spin correlation functions in B-spinel lattice

High-temperature series expansion of the spin correlation functions on the B-spinel lattice are computed to order 6 in for Heisenberg model having both nearest- and next-nearest-neighbour exchange integrals. The results are given for various neighbour correlations (up to the third). The behaviour with the temperature and the site dilution is presented. The obtained results provide a useful tool for a straightforward interpretation and understanding of experimental data. The approach is applied to the experimental results of the B-spinel in the dilution range . The critical temperature and the critical exponents for the susceptibility and the correlation length are deduced by applying the Padé approximant methods. The following estimates are obtained for the familiar critical exponents: and . These values are not sensitive to the dilution ratio x. The transition temperatures as a function of x obtained by the present theory are found to be in excellent agreement with the experimental ones.

Note on the Jacobi Elliptic Functions

An algebraic method is given for the power series expansions of the Jacobi elliptic functions, based on the properties of the algebra associated with differential equations defining the functions.

A Polynomial Time Complexity Bound for Computations on Curves

In this paper, we use a recent quantitative version of Eisenstein's theorem on power series expansions of algebraic functions to compute a polynomial-time bit complexity bound for the computation of the genus of an algebraic curve over the rationals. The result is extendable to any field of characteristic zero which is finitely generated over the rationals.

Singularity Analysis and Boundary Integral Equation Method for Frictional Crack Problems in Two-Dimensional Elasticity

This study investigates the stress singularities in the neighborhood of the tip of a sliding crack with Coulomb-type frictional contact surfaces, and applies the boundary integral equation method to solve some frictional crack problems in plane elasticity. A universal approach to the determination of the complex order of stress singularity is established analytically by using the series expansion of the complex stress functions. When the cracks are open, or when no friction exists between the upper and lower crack faces, our results agree with those given by Williams. When displacement and traction are prescribed on the upper and lower crack surfaces (or vice versa), our result agrees with those by Muskhelishvili. For the case of a closed crack with frictional contact, the only nonzero stress intensity factor is that for pure shear or sliding mode. By using the boundary integral equation method, we derive analytically that the stress intensity factor due to the interaction of two colinear frictional cracks under far field biaxial compression can be expressed in terms of E(k) and K(k) (the complete elliptic integrals of the first and second kinds), where k=[1-(a/b)2]1/2 with 2a the distance between the two inner crack tips and b- a the length of the cracks. For the case of an infinite periodic colinear crack array under remote biaxial compression, the mode II stress intensity factor is found to be proportional to [2b tan(π a/2b)]1/2 where 2a and 2b are the crack length and period of the crack array.

Squeezing Properties of the Ion Traps

With time and space transformation, we first solve the Schrödinger equation of the time-dependent harmonic oscillator (TDHO) system. The properties of the squeezing in the case of , i.e., the Paul trap and , namely a stable frequency interfered by a single-pulsing-type disturbance, are investigated by using function series expansion. Results in the former case are compared with the numerical calculations carried out before. It is found that the condition related to the most strong squeezing is the direct result of our method, not any approximation is needed. Solutions of the latter one are valuable for the study of the ion trap, where different cases are studied in detail. Finally we made a discussion for the measurement and observation of a trapped ion by using squeezing properties.

GENERALISED TRANSFER FUNCTIONS OF NEURAL NETWORKS

When artificial neural networks are used to model non-linear dynamical systems, the system structure which can be extremely useful for analysis and design, is buried within the network architecture. In this paper, explicit expressions for the frequency response or generalised transfer functions of both feedforward and recurrent neural networks are derived in terms of the network weights. The derivation of the algorithm is established on the basis of the Taylor series expansion of the activation functions used in a particular neural network. This leads to a representation which is equivalent to the non-linear recursive polynomial model and enables the derivation of the transfer functions to be based on the harmonic expansion method. By mapping the neural network into the frequency domain information about the structure of the underlying non-linear system can be recovered. Numerical examples are included to demonstrate the application of the new algorithm. These examples show that the frequency response functions appear to be highly sensitive to the network topology and training, and that the time domain properties fail to reveal deficiencies in the trained network structure.

A new functional expansion for nonlinear systems

In this paper, we shall introduce a new functional series expansion of the output of a general nonlinear system as power series in powers of derivatives of the input components. We shall illustrate the usefulness of this series by considering two important problems namely those of “realization” and systems inversion.

Goodman's conjectures on the coefficients of multivalent functions

The purpose of this paper is lo survey known results on two conjectures made by A. W. Goodman concerning the coefficients in the power series expansions of multivalent functions.

Approximate study of the quantum statistics of two Paul-trapped ions

Using an earlier assumption, the quantum properties of two Paul-trapped ions along an axis are further investigated in this work. The validity of the approximation is first tested by function series expansion (FSE). Then it is interestingly found in the discussion of the quantum statistics that the stable confinement of two light Paul-trapped ions is determined not only by the condition of the stability region, but also by the charges and spins of the two ions.

Modeling noise and modulation performance of fiber grating external cavity lasers

A detailed model is developed for analyzing fiber grating external cavity lasers for both static and small-signal modulation conditions. The chip and package parasitics and leakage current induced distortion are included. The composite system is solved analytically in the small-signal regime using a Volterra functional series expansion method. As an application of the model, a thorough analysis of the appearance of nulls close to the harmonics of the cavity resonance frequency in the noise and modulation spectra is given. We show that the appearance of these nulls can be explained using the interplay of amplitude and phase coupling between laser diode and external resonant cavity. A signal flow graph approach is introduced which identifies methods of minimizing the nulls.

Modeling stochastic spike train responses of neurons: an extended Wiener series analysis of pigeon auditory nerve fibers.

A general method for developing data-based, stochastic nonlinear models of neurons by means of extended functional series expansions was applied to neural activities of pigeon auditory nerve fibers responding to Gaussaian white noise stimuli. To determine Wiener series representations of the investigated neurons the fast orthogonal search algorithm was used. The results suggest that nonlinearities are only instantaneous and that the signal transduction of the investigated sensory system can be described by cascades of dynamic linear and static nonlinear devices. However, only slight improvements result from the nonlinear terms. Considerable improvements are, nevertheless, possible by generalizing the ordinary Wiener series, so that prior neural activity can be taken into account. These extended series were used to develop stochastic models of spiking neurons. The models are able to generate realistic interspike interval distributions and rate-intensity functions. Finally, it will be shown that the irregularity in real and modeled action potential trains has advantages concerning the decoding of neural responses.

A COMPLEX VARIABLE SOLUTION FOR A DEFORMING CIRCULAR TUNNEL IN AN ELASTIC HALF‐PLANE

An analytical solution is presented of problems for an elastic half-plane with a circular tunnel, which undergoes a certain given deformation. The solution uses complex variables, with a conformal mapping onto a circular ring. The coefficients in the Laurent series expansion of the stress functions are determined by a combination of analytical and numerical computations. As an example the case of a uniform radial displacement of the tunnel boundary is considered in some detail. It appears that a uniform radial displacement is accompanied by a downward displacement of the tunnel as a whole. This phenomenon also means that the distribution of the apparent spring constant is strongly non-uniform. © 1997 by John Wiley & Sons, Ltd.

A COMPLEX VARIABLE SOLUTION FOR A DEFORMING CIRCULAR TUNNEL IN AN ELASTIC HALF-PLANE

SUMMARY An analytical solution is presented of problems for an elastic half-plane with a circular tunnel, whichundergoesacertaingivendeformation.Thesolutionusescomplexvariables,withaconformalmappingontoa circular ring. The coeƒcients in the Laurent series expansion of the stress functions are determined bya combination of analytical and numerical computations. As an example the case of a uniform radialdisplacement of the tunnel boundary is considered in some detail. It appears that a uniform radial displace-mentisaccompaniedbyadownwarddisplacementofthetunnelasawhole.Thisphenomenonalsomeansthatthedistribution oftheapparentspring constantis strongly non-uniform. ( 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods geomech., vol. 21, 77 — 89 (1997)(No. of Figures: 7 No. of Tables: 1 No. of Refs: 10) Key words: elasticity; tunnel; complex variables INTRODUCTIONIn this paper the stresses and displacements in an elastic half-plane due to the deformation of acircular tunnel are considered. The method used is the complex variable method. 1 The boundaryconditions arethat the upper boundaryof the half-planeisfreeof stress, andthat at theboundaryofthetunnelthedisplacementisprescribed.Thisisusuallycalledthesecondtypeofboundarycondition.In order to solve the problem, a conformal transformation onto a circular ring is used, and in thetransformed plane the complex stress functions are represented by their Laurent series expansions.IntheclassicaltreatisesofMuskhelishvili1andSokoliniko⁄2ontheapplicationofthecomplexvariable method in elasticity, the class of problems studied here, involving a multiply connectedregion and conformal mapping onto a circular ring, is briesy mentioned, but it is stated thatOdiƒcultiesO arise in the solution of these problems, and it is suggested to use another method ofsolution, such as the method using bipolar co-ordinates.3