Bounds and algorithms for the -Bessel function of imaginary order

Booker, Andrew R.; Strömbergsson, Andreas; Then, Holger

doi:10.1112/s1461157013000028

Cited by 13 publications

(12 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and the known bounds for this latter function (see, e.g., Booker et al [2]). Since for 0 < λ < 1 we do not have an inequality like (3.4), it seems hard to obtain any usable error bound which is appropriate when arg ν is close to ± π 2 .…”

Section: Error Boundsmentioning

confidence: 90%

The resurgence properties of the large order asymptotics of the Anger-Weber function II

Nemes¹

2014

Journal of Classical Analysis

View full text Add to dashboard Cite

Abstract. In this paper, we derive a new representation for the Anger-Weber function, employing the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373-396). As a consequence of this representation, we deduce a number of properties of the large order asymptotic expansion of the Anger-Weber function, including explicit and realistic error bounds, asymptotic approximations for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. IntroductionIn the first part of this series of papers [7], we proved new resurgence-type representations for the remainder term of the asymptotic expansion of the Anger-Weber function A −ν (λν) with complex ν and λ ≥ 1. These resurgence formulas have different forms according to whether λ > 1 or λ = 1. The main goal of this paper is to derive a similar representation for the Anger-Weber function A ν (λν) with complex ν and λ > 0. Our derivation is based on the reformulation of the method of steepest descents by Howls [4]. Using this representation, we obtain a number of properties of the large order asymptotic expansion of the Anger-Weber function, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.Our first theorem describes the resurgence properties of the asymptotic expansion of A ν (λν) for λ > 0. The notations follow the ones given in [11, p. 298]. Throughout this paper, empty sums are taken to be zero. Theorem 1.1. Let λ > 0 be a fixed positive real number, and let N be a non-negative integer. Then we have it (λit) dt.

show abstract

Section: Error Boundsmentioning

confidence: 90%

The resurgence properties of the large order asymptotics of the Anger-Weber function II

Nemes¹

2014

Journal of Classical Analysis

View full text Add to dashboard Cite

show abstract

“…Let us now consider the case θ = π/2 (or, equivalently, t = y). Apply [21, p. 78 (8) and p. 247 (5)] to obtain…”

Section: First Case: Y ≥ T ≥mentioning

confidence: 99%

“…Since we only need a bound for small y, it suffices to adapt the bound for purely imaginary order from [5,Sect. 3.1].…”

Section: Proof Of Proposition 18mentioning

confidence: 99%

Eisenstein series and an asymptotic for the K-Bessel function

Tseng

2021

Ramanujan J

View full text Add to dashboard Cite

We produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

show abstract

“…As we show in §2, this can be done, and the two are related essentially by the factor (1.1).We analyze this strategy in greater detail in §2, but the upshot is that to compute Maass form L-functions for a wide range of values of r and t, it suffices to compute f (ie iθ u) for suitable values of θ and u. In turn, using modularity to move each point to the fundamental domain, the problem reduces to computing the K-Bessel function K ir (y) for various r and y. Fortunately, that is a problem that underlies all computational aspects of Maass cusp forms and has been well studied; see, for instance, [3].…”

mentioning

confidence: 99%

Rapid computation of L-functions attached to Maass forms

Booker¹,

Then²

2018

Int. J. Number Theory

Self Cite

View full text Add to dashboard Cite

Let L be a degree-2 L-function associated to a Maass cusp form. We explore an algorithm that evaluates t values of L on the critical line in time O(t 1+ε ). We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution.Thus, Rubinstein's exponential factor arises naturally from a contour rotation.For a Maass form f of weight 0 and even parity (say), we similarly have the integral representationSince f is no longer holomorphic in this case, we cannot simply rotate the contour in this expression, but we are free to start with the rotated integral ∞ 0 f (ie iθ u)u s− 1 2 du u and try to relate it back to Λ(s). As we show in §2, this can be done, and the two are related essentially by the factor (1.1).We analyze this strategy in greater detail in §2, but the upshot is that to compute Maass form L-functions for a wide range of values of r and t, it suffices to compute f (ie iθ u) for suitable values of θ and u. In turn, using modularity to move each point to the fundamental domain, the problem reduces to computing the K-Bessel function K ir (y) for various r and y. Fortunately, that is a problem that underlies all computational aspects of Maass cusp forms and has been well studied; see, for instance, [3].

show abstract

Bounds and algorithms for the -Bessel function of imaginary order

Cited by 13 publications

References 35 publications

The resurgence properties of the large order asymptotics of the Anger-Weber function II

The resurgence properties of the large order asymptotics of the Anger-Weber function II

Eisenstein series and an asymptotic for the K-Bessel function

Rapid computation of L-functions attached to Maass forms

Contact Info

Product

Resources

About