The evaluation of Bessel functions via exp-arc integrals

Borwein, David; Borwein, Jonathan M.; Chan, O-Yeat

doi:10.1016/j.jmaa.2007.10.003

Cited by 7 publications

(12 citation statements)

References 13 publications

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“…This shows that by using non-maximal intervals both series in (2.7) now converge at a much faster, geometric rate; this has been seen in [4,5,11]. In addition, both series in (2. .…”

Section: The Modified Bessel Function I ν (Z)mentioning

confidence: 69%

“…In this section we derive error bounds on the tails of the Hadamard series encountered in Sections 3 and 4 when they are truncated after M terms; analogous bounds have been given in [5]. Consider the tails …”

Section: Bounds On the Tails Of Hadamard Seriesmentioning

confidence: 96%

“…It is also an obvious consequence of our path decomposition in (4.2) that truncation of the sum over n in (4.9) after N terms yields an error of O(e −Ω N x ). It is worth pointing out that, instead of a truncation of the tail of the integral, it is possible to apply an expansion at infinity as described in [5].…”

Section: The Expansion For U (A a + B Z)mentioning

confidence: 99%

“…In this section we discuss the theory developed in [4,5] Making the change of variable u = sin 1 2 t and employing the series expansion of the factor 9 exp(−2im arcsin u), they obtain…”

Section: Connection With the Work Of Borwein Et Al In [45]mentioning

confidence: 99%

“…In this way the Hadamard expansion in (1.3) becomes an interesting alternative in high-precision evaluation. A different procedure applied to the Bessel functions using the so-called 'exp-arc' integrals has recently been given in [4,5]. This latter mode of expansion results in series that converge at a much faster (geometric) rate, with the late terms for I ν (z) being of O(2 −k k −ν−3/2 ).…”

Section: Introductionmentioning

confidence: 99%

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High-precision evaluation of the Bessel functions via Hadamard series

Paris¹

2009

Journal of Computational and Applied Mathematics

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a b s t r a c tWe present a method of high-precision calculation of the Bessel functions using Hadamard series. Such series are absolutely convergent expansions involving the normalised incomplete gamma function P(a, z) = γ (a, z)/ (a) and possess early terms that behave like those in an asymptotic expansion. In the case of real variables the function P(a, z) acts as a smoothing factor on the terms of the series. We show how these series representing the Bessel functions of complex argument can be chosen so as to produce rapidly convergent series that possess terms decaying at the geometric rate ϑ k , where 0 < ϑ < 1 and k is the ordinal number of the series. We give numerical examples with ϑ = 1 2 , 1 3 and 1 4 . The theory is extended to cover the confluent hypergeometric functions 1 F 1 (a; b; z) and U(a, b, z), thereby dealing with many of the special functions arising in mathematical physics.

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“…This shows that by using non-maximal intervals both series in (2.7) now converge at a much faster, geometric rate; this has been seen in [4,5,11]. In addition, both series in (2. .…”

Section: The Modified Bessel Function I ν (Z)mentioning

confidence: 69%

Section: Bounds On the Tails Of Hadamard Seriesmentioning

confidence: 96%

Section: The Expansion For U (A a + B Z)mentioning

confidence: 99%

“…In this section we discuss the theory developed in [4,5] Making the change of variable u = sin 1 2 t and employing the series expansion of the factor 9 exp(−2im arcsin u), they obtain…”

Section: Connection With the Work Of Borwein Et Al In [45]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-precision evaluation of the Bessel functions via Hadamard series

Paris¹

2009

Journal of Computational and Applied Mathematics

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High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion

Navas-Palencia

2017

Adv Comput Math

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We examine the use of the Euler-Maclaurin formula and new derived uniform asymptotic expansions for the numerical evaluation of the Lerch transcendent Φ(z, s, a) for z, s, a ∈ C to arbitrary precision. A detailed analysis of these expansions is accompanied by rigorous error bounds. A complete scheme of computation for large and small values of the parameters and argument is described along with algorithmic details to achieve high performance. The described algorithm has been extensively tested in different regimes of the parameters and compared with current state-of-the-art codes. An open source implementation of Φ(z, s, a) based on the algorithms described in this paper is available.

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On a novel iterative method to compute polynomial approximations to Bessel functions of the first kind and its connection to the solution of fractional diffusion/diffusion-wave problems

Yuste¹,

Abad²

2011

J. Phys. A: Math. Theor.

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Abstract. We present an iterative method to obtain approximations to Bessel functions of the first kind J p (x) (p > −1) via the repeated application of an integral operator to an initial seed function f 0 (x). The class of seed functions f 0 (x) leading to sets of increasingly accurate approximations f n (x) is considerably large and includes any polynomial. When the operator is applied once to a polynomial of degree s, it yields a polynomial of degree s + 2, and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree. We focus on the set of polynomial approximations generated from the seed function f 0 (x) = 1. This set of polynomials is not only useful for the computation of J p (x), but also from a physical point of view, as it describes the long-time decay modes of certain fractional diffusion and diffusion-wave problems.

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The evaluation of Bessel functions via exp-arc integrals

Cited by 7 publications

References 13 publications

High-precision evaluation of the Bessel functions via Hadamard series

High-precision evaluation of the Bessel functions via Hadamard series

High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion

On a novel iterative method to compute polynomial approximations to Bessel functions of the first kind and its connection to the solution of fractional diffusion/diffusion-wave problems

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