Bounds and algorithms for the K-Bessel function of imaginary order

Andrew R Booker; Andreas Strömbergsson; Holger L Then

doi:10.1112/S1461157013000028

Bounds and algorithms for the K-Bessel function of imaginary order

Andrew R Booker, Andreas Strömbergsson, Holger L Then

Research output: Contribution to journal › Article (Academic Journal) › peer-review

8 Citations (Scopus)

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Abstract

Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of . Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of .

Original language	English
Pages (from-to)	78-108
Number of pages	31
Journal	LMS Journal of Computation and Mathematics
Volume	16
Early online date	10 Apr 2013
DOIs	https://doi.org/10.1112/S1461157013000028
Publication status	Published - 2013

Keywords

Bessel function

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10.1112/S1461157013000028

Full-text PDF (accepted author manuscript)
This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Cambridge University Press at https://www.cambridge.org/core/journals/lms-journal-of-computation-and-mathematics/article/bounds-and-algorithms-for-the-kbessel-function-of-imaginary-order/AB68D056F1A8398C9E06C61E2D94529A. Please refer to any applicable terms of use of the publisher.
Accepted author manuscript, 454 KB

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Explicit number theory, automorphic forms and L-functions
Booker, A. R. (Principal Investigator)
1/10/09 → 1/04/15
Project: Research

Cite this

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AU - Strömbergsson, Andreas

AU - Then, Holger L

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N2 - Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of . Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of .

AB - Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of . Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of .

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DO - 10.1112/S1461157013000028

M3 - Article (Academic Journal)

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SP - 78

EP - 108

JO - LMS Journal of Computation and Mathematics

JF - LMS Journal of Computation and Mathematics

ER -

Bounds and algorithms for the K-Bessel function of imaginary order

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Keywords

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Projects

Explicit number theory, automorphic forms and L-functions

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