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Computation of Hyperspherical Bessel Functions

Part of: Difference and functional equations, recurrence relations

Published online by Cambridge University Press:  06 July 2017

Thomas Tram
Thomas Tram*
Affiliation:
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark
*
*Corresponding author. Email address:thomas.tram@phys.au.dk (T. Tram)
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Abstract

In this paper we present a fast and accurate numerical algorithm for the computation of hyperspherical Bessel functions of large order and real arguments. For the hyperspherical Bessel functions of closed type, no stable algorithm existed so far due to the lack of a backwards recurrence. We solved this problem by establishing a relation to Gegenbauer polynomials. All our algorithms are written in C and are publicly available at Github [https://github.com/lesgourg/class_public]. A Python wrapper is available upon request.

Keywords

Hyperspherical Bessel functionsrecurrence relationsWKB approximationinterpolation

MSC classification

Secondary: 65Q30: Recurrence relations 65Z05: Applications to physics
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 852 - 862
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Jie Shen

References

[1] Polyanin, A. D. and Zaitsev, V. F., Handbook of exact solutions for ordinary differential equations. CRC Press, Boca Raton, 1995.Google Scholar
[2] Harrison, E. R., Normal modes of vibrations of the universe, Rev. Mod. Phys. 39 (Oct, 1967) 862–882.Google Scholar
[3] Abbott, L. and Schaefer, R. K., A General, Gauge Invariant Analysis of the Cosmic Microwave Anisotropy, Astrophys. J. 308 (1986) 546.Google Scholar
[4] Kosowsky, A., Efficient computation of hyperspherical bessel functions, astro-ph/9805173.Google Scholar
[5] Johansson, F. et al., mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.18), December, 2013.Google Scholar
[6] Gautschi, W., Computational aspects of three-term recurrence relations, SIAM Review 9 (Jan., 1967) 24–82.Google Scholar
[7] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical recipes in C (2nd ed.): the art of scientific computing. Cambridge University Press, New York, NY, USA, 1992.Google Scholar
[8] Lentz, W. J., Generating bessel functions in mie scattering calculations using continued fractions, Appl. Opt. 15 (Mar, 1976) 668–671.Google Scholar
[9] Thompson, I. and Barnett, A., Coulomb and bessel functions of complex arguments and order, Journal of Computational Physics 64 (1986) 490–509.Google Scholar
[10] Langer, R. E., On the asymptotic solutions of ordinary differential equations, with reference to the stokes’ phenomenon about a singular point, Transactions of the American Mathematical Society 37 (1935) pp. 397–416.Google Scholar
[11] Bender, C. and Orszag, S., Advanced mathematical methods for scientists and engineers. International series in pure and applied mathematics. McGraw-Hill, 1978.Google Scholar