Exactification of the asymptotics for Bessel and Hankel functions
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, Jν(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.
Section snippets
Complete asymptotic expansion for Bessel functions
In order to derive a complete asymptotic expansion for the general Bessel function Jν(z) where ν is any number in the complex plane, we shall employ the asymptotic forms for hypergeometric functions given in Luke [4]. In the present work we shall only consider positive real values of the variable z, leaving the complex-valued case for a future publication. To employ the results in [4], we need to transform Jν(z) in the form of a hypergeometric function. This is accomplished by using the power
Mellin–Barnes regularization of divergent series
Since Eq. (13) is a complete asymptotic expansion, it is composed of divergent series. However, in this section we present the technique of Mellin–Barnes regularization of a divergent series, which seeks to express each divergent series in Eq. (13) in terms of a Mellin–Barnes integral that can be evaluated numerically. As shown in [3], [13] the technique is equivalent to Borel summation, but in many instances it is more expedient from a computational view than the latter.
Mellin–Barnes
Borel summation of the Bessel asymptotic series
To obtain the Borel-summed form of the asymptotic expansion for Bessel functions, we shall introduce the following identity in Eq. (10):where B(x,y) denotes the beta function. Then the gamma and beta functions are replaced by their integral representations, viz.Thus, Eq. (10) becomes
Numerical results
In this section we aim to use the asymptotic forms of the previous sections to obtain exact values of the Bessel function for various values of the order ν, but not large |ν|, and for various real values of z. In so doing, we shall see that a complete asymptotic expansion is merely another representation of the original function, albeit with special properties.
The special properties relate to the behaviour of the remainder. As z approaches infinity, the limiting point of the asymptotic forms of
Conclusion
In this paper we have seen for positive real values of the variable z and for fixed values of ν that there are no subdominant exponential asymptotic series in the most commonly used asymptotic expansion for Bessel functions. Hence, , , represent complete asymptotic expansions for Bessel and Hankel functions. Via these results Borel-summed versions in the form of convergent two-dimensional integral representations have been developed for the remainders of these asymptotic expansions. For the
Acknowledgements
The author wishes to thank Dr. Andrew A. Rawlinson, University of Melbourne, for advice and assistance on numerical aspects of this work.
References (19)
Asymptotic Expansions: Their Derivation and Interpretation
(1973)- et al.
Generalised Euler–Jacobi Inversion Formula and Asymptotics beyond All Orders, London Mathematical Society Lecture Note, vol. 214
(1995) The Special Functions and Their Approximations
(1969)- et al.
Hyperasymptotics
Proc. Roy. Soc. London A
(1990) - et al.
Hyperasymptotics for integrals with saddle points
Proc. Roy. Soc. London A
(1991) - et al.
Hyperasymptotic solutions of second-order ordinary differential equations I
Methods Appl. Anal.
(1995) - R.B. Paris, Hyperasymptotics and Mellin–Barnes Integrals, Technical Report MS(98:01), University of Abertay, UK,...
A recursion formula for the coefficients in an asymptotic formula
Proc. Glasgow Math. Assoc.
(1958)
Cited by (15)
Landau damping, Stokes phenomenon and the weakly magnetized Maxwellian plasma
2005, Physics Letters, Section A: General, Atomic and Solid State PhysicsExact values of the gamma function from Stirling's formula
2020, MathematicsDivide-and-conquer approach applied to impedance analysis of pipe eddy current test
2016, Dianzi Keji Daxue Xuebao/Journal of the University of Electronic Science and Technology of China