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

Part of: Inequalities Functional analytic methods in summability Numerical approximation and computational geometry Functions of several variables Stability theory Approximations and expansions Computational aspects

Published online by Cambridge University Press:  10 April 2013

Andrew R. Booker ,
Andreas Strömbergsson  and
Holger Then
Andrew R. Booker
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom email andrew.booker@bristol.ac.uk
Andreas Strömbergsson
Affiliation:
Department of Mathematics, Box 480, Uppsala University, S-75106 Uppsala, Sweden email andreas.strombergsson@math.uu.se
Holger Then
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom email holger.then@bristol.ac.uk

Abstract

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Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function ${K}_{ir} (x)$ 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 ${K}_{ir} (x)$ and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of $r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of ${K}_{ir} (x)$.

MSC classification

Secondary: 26D07: Inequalities involving other types of functions 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 33F05: Numerical approximation and evaluation 34D05: Asymptotic properties 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 41A80: Remainders in approximation formulas 65D05: Interpolation 40H05: Functional analytic methods in summability 26B99: None of the above, but in this section
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 78 - 108
Copyright
© The Author(s) 2013 

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