Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

Calculation of Bessel functions by using continued fractions

We propose a new method for the calculation of Bessel functions of the first kind of integral order. By using the Laplace transformation, we solve a linear differential equation that defines the generating function for the Bessel functions expressed in terms of continued fractions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1704–1705, December, 1995.     

Stress concentration in a three-layer plate with two cylindrical cavities

The stressed state of a thick three-layer plate with two circular holes is studied. Resolving functions are expressed in terms of Kolosov-Muskhelishvili integral potentials and modified Bessel functions. The problem is reduced to solving an infinite set of linear algebraic equations.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 25–28, 1987.     

New formulae for higher order derivatives and applications

We present new formulae (the Slevinsky–Safouhi formulae I and II) for the analytical development of higher order derivatives. These formulae, which are analytic and exact, represent the kth derivative as a discrete sum of only k+1 terms. Involved in the expression for the kth derivative are coefficients of the terms in the summation. These coefficients can be computed recursively and they are not subject to any computational instability. As examples of applications, we develop higher order derivatives of Legendre functions, Chebyshev polynomials of the first kind, Hermite functions and Bessel functions. We also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations. We also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical Bessel integral functions.     

Implementation of Miller's method for evaluation of Bessel functions of first kind

The implementation of Miller's iterative method is considered. A number of techniques are proposed, extending the applicability of the method to various computing systems. We discuss the evaluation of Bessel functions of first kind by these techniques in a wide range of real argument and index values and also for a number of complex argument values and an arbitrary real index.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 65–69, 1987.     

Asymptotic solution of a problem with one and two cusp points

The perturbation method and the Langer transform method are applied to obtain a second approximate solution of the problem with one and two cusp points. As an example, Langer type asymptotic formulas are derived for Bessel functions.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 42–50, 1989.     

On the modified Taylor's approximation for the solution of linear and nonlinear equations

Fourier series and Taylor's expansions are commonly used in the fields of science and engineering. A whole range of interesting physical problems can be solved using one or the other of these standard techniques. Yet, there is a class of problems whose solutions exhibit near sinusoidal or repetitive behavior that cannot be solved using the above expansions. It is for these types of problems that the modified Taylor expansion has been developed. It is a method that combines the advantages of the repetitive behavior of sinusoidal functions and polynomial series. This technique has not been employed for tackling equations whose solutions are near sinusoidal. We discuss the method and apply it to a number of interesting problems that show its utility. Examples include the solution of differential, integral, integro-differential and functional equations. In addition, we propose an algorithm for the numerical integration of a function which exhibits near periodic behavior, namely the Bessel function. Most of the symbolic and numerical computations have been performed using the Computer Algebra System––Maple.     

Simple analysis of a fluid queue driven by an M/M/1 queue

In this note we consider the fluid queue driven by anM/M/1 queue as analysed by Virtamo and Norros [Queueing Systems 16 (1994) 373–386]. We show that the stationary buffer content in this model can be easily analysed by looking at embedded time points. This approach gives the stationary buffer content distribution in terms of the modified Bessel function of the first kind of order one. By using a suitable integral representation for this Bessel function we show that our results coincide with the ones of Virtamo and Norros.     

Solution of the 2D ising model on a triangular lattice by the method of auxiliaryq-deformed Grassmann fields

A representation in the form of a functional integral is obtained for the partition function of the inhomogeneous 2D Ising model on a triangular lattice where the coupling parameters are arbitrary functions of coordinates. The method for transforming the partition function into an integral uses an auxiliary six-component Grassmann field in which the Grassmann fields corresponding to one of the components commute with the others. Thus, one pair of components realizes a representation of the q-deformed group SL_q(2, R) with q=–1 and the other two pairs correspond to the usual Grassmann spinors (q=1). An explicit expression in terms of the modified Pfaffian is found for the Gaussian integral over these fields and its relation to the ordinary Grassmann functional integral is established.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 441–463, December, 1996.     

A trivariate law for certain processes related to perturbed Brownian motions

D. Williams' path decomposition and Pitman's representation theorem for BES(3) are expressions of some deep relations between reflecting Brownian motion and the 3-dimensional Bessel process.In [Ph. Carmona et al., Stochastic Process. Appl. 7 (1999) 323–333], we presented an attempt to relate better reflecting Brownian motion and the 2-dimensional Bessel process, using space and time changes related to the Ray–Knight theorems on local times, in the manner of Jeulin [Lect. Notes Math., vol. 1118, Springer, Berlin, 1985] and Biane–Yor [Bull. Sci. Math. 2ème Sér. 111 (1987) 23–101].Here, we characterize the law of a triplet linked to the perturbed Brownian motion which naturally arises in [Ph. Carmona et al., Stochastic Proc. Appl. 7 (1999) 323–333], and we point out its relations with Bessel processes of several dimensions.The results provide some new understanding of the generalizations of Lévy's arc sine law for perturbed Brownian motions previously obtained by the second author.     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

A generalization of classical orthogonal polynomials and the convergence of simultaneous Pade approximants

The concept of generalized classical polyorthogonal polynomials and, in particular, that of generalized Laguerre polynomials, corresponding to a collection of measures with supports on infinite rays in the complex plane, is introduced. The asymptotic behavior of these polynomials and of their corresponding functions of the second kind is investigated. Moreover, generalizations of the Bessel functions and of the Euler integral of the second kind are defined and investigated. The convergence of the simultaneous Pade approximants to certain Stieltjes type functions is proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 125–165, 1986.     

Multiplier family of harmonic univalent functions

The aim of this paper is to study a multiplier family of harmonic univalent functions using the sequences {c_n} and {d_n} of positive real numbers. By specializing {c_n} and {d_n}, the generalized Bernardi–Libera–Livingston integral operator is modified for such functions and the closure of the multiplier family under the modified integral operator is determined. Also, convolution products, closure properties, distortion theorems, convex combinations and neighborhoods for such functions are given.     

On a space of potentials of fractional smoothness with densities from the stepanov class

We introduce a new functional space of functions of several variables of fractional smoothness admitting integral representations in the form of the convolution with the Bessel-Macdonald matrix kernel and having densities from the Stepanov class. The Liouville-Sobolev space (the space of Bessel potentials) is imbedded in this space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1399–1404, October, 1995.     

Distribution of integral functionals of the local time of a Bessel process

In the paper one considers methods allowing one to determine the distribution of certain functionals of the local times of diffusion processes. These methods turn out to be most effective for the Brownian local time and for the local time of a third-order Bessel process. For Brownian local time these methods have been obtained earlier and the main part of the paper is devoted to the development of methods for the evaluation of the distributions of integral functionals and functionals of the supremum type of the local time of a third-order Bessel process.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 8–27, 1989.     

The evaluation of Bessel functions via exp-arc integrals

A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J.M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an “exp-arc” integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.     

Basics of diamond- partial dynamic calculus on time scales

In this paper we introduce partial diamond-α dynamic derivatives for two-variable functions and the double integral calculus via the diamond-α dynamic integral on time scales. Also we establish a two-dimensional weighted Hardy–Knopp type integral inequality on time scales.     

On the optimization of direct methods for solving fredholm integral equations of the second kind with infinitely smooth kernels

We give a direct method, optimal inL ₂, for solving the Fredholm integral equation of the second kind with operators acting into the space of functions harmonic in a disk or into the space of functions that can be analytically extended to an infinite strip. The exact order of the error of this method is determined.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1695–1701, December, 1993.     

Bayes estimation in a mixture inverse Gaussian model

In this paper a mixture model involving the inverse Gaussian distribution and its length biased version is studied from a Bayesian view-point. Using proper priors, the Bayes estimates of the parameters of the model are derived and the results are applied on the aircraft data of Proschan (1963,Technometrics,5, 375–383). The posterior distributions of the parameters are expressed in terms of the confluent-hypergeometric function and the modified Bessel function of the third kind. The integral involved in the expression of the estimate of the mean is evaluated by numerical techniques.     

Quasistatic and dynamic problems in thermoelasticity for infinite two-layer rods

The method of influence functions is used to solve problems concerning the structure of a nonstationary temperature field and the associated quasistatic and dynamic displacement and stress fields in infinite two-layer rods for the most general assumptions in the classical theory of thermoelasticity. For constructing the influence functions we apply the method of integral Fourier transforms on a line with one conjugate point and an operational method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 78–86, 1990.