Exactification of the asymptotics for Bessel and Hankel functions

Abstract

Exactification is the process of showing how a complete asymptotic expansion can be evaluated to yield exact values of the original function it represents. Because the process does not involve neglecting the remainders of truncated component asymptotic series within a complete asymptotic expansion, techniques for evaluating divergent series are required. One such technique is Borel summation, but in many instances, it can be computationally slow and consequently, lacking in precision. Another is the numerical technique of Mellin–Barnes regularization, which is capable of evaluating divergent series with great precision far more rapidly than Borel summation. Here, both techniques are presented in the evaluation of exact values for Bessel and Hankel functions from their complete asymptotic expansions.

Introduction

A complete asymptotic expansion [1] may be defined as an expansion that not only contains all the terms of a dominant asymptotic series, but also all the terms of subdominant asymptotic series, where they exist. In some cases there may be more than one of these subdominant series appearing in the expansion. Generally, when mathematicians are required to derive asymptotic expansions for functions or integrals, they are only interested in obtaining the first few leading order terms of the dominant series because the higher-order terms become divergent. As a consequence, the subdominant exponential terms, which are said to lie beyond all orders in a complete asymptotic expansion [2], are also neglected, despite the fact that they too become divergent.

Whenever a complete asymptotic expansion is truncated, information is lost and there is, of course, no possibility that the truncated part can ever yield exact values of the function or integral it represents. Thus, the subject of asymptotics has long been regarded as a discipline that suffers from the drawbacks of vagueness and limited range of applicability [1]. This view, however, has been changed recently as a result of the work done by Kowalenko et al. [3].

In their study of the exponential series, S₃(a)=∑_n=0^∞exp(−an³), these authors were able to determine a complete asymptotic expansion for a→0, which consisted of a divergent algebraic series and a divergent subdominant exponential series. By applying the novel numerical technique of Mellin–Barnes regularization to both component series they found to arbitrary accuracy, in some cases as high as 63 significant figures, that they could exactify these functions in regions where the asymptotic expansion was previously thought to be inapplicable, viz. for values of a much greater than unity. Exactification is defined here as the process of calculating both the truncated sum of a divergent series and the divergent remainder/tail of the series such that when both entities are combined, they yield the exact value of the original function. Thus, by applying Mellin–Barnes regularization to the complete asymptotic expansion of S₃(a), Kowalenko et al. were able to evaluate S₃(10) exactly to 15 decimal places. Greater accuracy could be obtained, but at the expense of more computer time.

In this paper we aim to apply the ideas and techniques in [3] to a study of the most frequently used asymptotic expansion for Bessel functions, J_ν(z). In Section 2 the complete expansion is derived via the asymptotic theorems for hypergeometric functions for positive real values of z. In Section 3 we present the definition/theorem, which establishes the validity of the Mellin–Barnes regularization of a typical divergent series. As an example, we show how the technique can be applied to the simplest divergent series, the geometric series. The section concludes by examining how the technique is to be implemented for the complete asymptotic expansion for Bessel functions. In Section 4 we develop convergent integral representations for the remainders of the asymptotic forms of Bessel and Hankel functions via Borel summation. These two-dimensional integrals reduce to known one-dimensional integral representations for the Hankel functions when the remainders represent the entire asymptotic series. Although the Borel-summed remainders are equivalent to the integrals obtained via Mellin–Barnes regularization, they are not always as amenable to numerical evaluation as the latter. Therefore, Section 5 presents a numerical study of the Mellin–Barnes regularized forms for various values of z. In nearly all instances we obtain the exact values for the Bessel and Hankel functions to 15 significant figures, which represent the limiting precision on our workstation. In Section 6 we conclude by summarizing the main points of this work and describing future work.