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

Published online by Cambridge University Press:  10 April 2013

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 ${K}_{ir} (x)$.

Type
Research Article
Copyright
© The Author(s) 2013 

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