Elementary approximations for zeros of Bessel functions

Let j_vk, y_vk and c_vk denote the kth positive zeros of the Bessel functions J_v(x), Y_v(x) and of the general cylinder function C_v(x) = cos αJ_v(x)?sin αY_v(x), 0 ? α < π, respectively. In this paper we extend to c_vk, k = 2, 3,..., some linear inequalities presently known only for j_vk. In the case of the zeros y_vk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero j_v1 we compare the linear enequalities given in [9] with some other known inequalities.     

Polynomial series expansions for confluent and Gaussian hypergeometric functions

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions and for Gaussian hypergeometric functions are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions and , in particular, if the parameters are also viewed as variables.

     

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.     

On an identity for zeros of Bessel functions

In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.     

Maximal order for quadratures usingn evaluations

The problem is to find approximationsI (f; h) to the integralI(f; h)= ₀ ^h f. Such an approximation has local orderp ifI(f; h)–I (f; h)=O(h ^p ) ash0. Let(n) denote the maximal local order possible for a method usingn evaluations of a function or its derivatives. We show that (n)=2n+1 if the information used is Hermitian. This is conjectured to be true in general. The conjecture is established for all methods using three or fewer evaluations.This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N00014-76-C-0370, NR 044-422. Numerical results reported in this paper were obtained through the computing facilities of the University of Maryland.     

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L² restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.     

A weighted uniform -estimate of Bessel functions: A note on a paper of Guo

An improved Guo's uniform estimate of Bessel functions is shown by using a uniform pointwise bound of Barceló and Córdoba.

     

Comments on the theory of Bessel functions with more than one index

We present an outline of the theory of Bessel-like functions with more than one index and one or more variables. Their link with other types of functions is discussed and their use in applications is touched on.     

Automatic,guaranteed integration of analytic functions

Davis introduced a method for estimating linear functionals of analytic functions by using Cauchy's Integral Formula. This is used to construct methods for numerical integration which give rigorous error bounds. By combining these bounds with strategies for order and subinterval adaptation, a program is developed for automatic integration of analytic functions. Interval analysis is used to validate the bounds.     

On sums of roots of infinite order entire functions

A class of infinite order entire functions is considered. Estimates for sums of the roots are derived. These estimates supplement the Hadamard theorem. Moreover, we establish a new estimate for the counting function of the roots, which in appropriate situations can be more useful than the Jensen inequality.     

Note on the Dočev-Grosswald asymptotic series for generalized Bessel polynomials

The Do?ev-Grosswald asymptotic series for the generalized Bessel polynomials y_n(z; a, b) is extended to O(1/n⁴) relative accuracy. The differential equation of the asymptotic factor, derived from the differential equation for y_n(z; a, b), is the basis of a different and easier method that employs simple recurrence relations and much less algebra for obtaining the same series. This is applied to the important special case of a = 1 to obtain the asymptotic series to O(1/n¹¹) relative accuracy.     

A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

In this paper, a collocation method is given to solve singularly perturbated two‐point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.