Advertisement

Numerische Mathematik

, Volume 136, Issue 3, pp 679–702 | Cite as

Bessel phase functions: calculation and application

  • David E. HorsleyEmail author
Article

Abstract

The Bessel phase functions are used to represent the Bessel functions as a positive modulus and an oscillating trigonometric term. This decomposition can be used to aid root-finding of certain combinations of Bessel functions. In this article, we give some new properties of the modulus and phase functions and some asymptotic expansions derived from differential equation theory. We find a bound on the error of the first term of this asymptotic expansion and give a simple numerical method for refining this approximation via standard routines for the Bessel functions. We then show an application of the phase functions to the root finding problem for linear and cross-product combinations of Bessel functions. This method improves upon previous methods and allows the roots in ascending order of these functions to be calculated independently. We give some proofs of correctness and global convergence.

Mathematics Subject Classification

33C10 33F05 65H05 

Notes

Acknowledgements

I would like to thank Prof. L.K. Forbes for providing some helpful comments on the manuscript, as well as the two anonymous referees useful remarks and insights. I am particularly grateful to the referee who suggested the works of Segura. This work was supported by an Australian Postgraduate Award at the University of Tasmania.

References

  1. 1.
    NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/, Release 1.0.13 of 2016-09-09, online companion to [2]
  2. 2.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (Eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York, NY (2010) Print companion to [1]Google Scholar
  3. 3.
    McMahon, J.: On the roots of the Bessel and certain related functions. Ann. Math. 9(1), 23–30 (1894)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Segura, J.: A global Newton method for the zeros of cylinder functions. Numer. Algorithm 18(3), 259–276 (1998). doi: 10.1023/A:1019125616736 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sorolla, E., Mattes, M.: Globally convergent algorithm to find the zeros of the cross-product of Bessel functions. Int. Conf. Electromagn. Adv. Appl. 1(3), 291–294 (2011). doi: 10.1109/ICEAA.2011.6046305 Google Scholar
  6. 6.
    Segura, J.: Computing the complex zeros of special functions. Numer. Math. 124(4), 723–752 (2013). doi: 10.1007/s00211-013-0528-6 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Watson, G.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (1966)zbMATHGoogle Scholar
  8. 8.
    Hankel, H.: Cylinderfunktionen erster und zweiter Art. Math. Ann. 1(3), 467–501 (1869)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)zbMATHGoogle Scholar
  10. 10.
    Olver, F.W.J.: The asymptotic expansion of Bessel functions of large order. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 247(930), 328–368 (1954)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Amos, D.E.: Algorithm 644: a portable package for bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Softw. 12(3), 265–273 (1986). doi: 10.1145/7921.214331 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Horsley, D.E.: Specialphase: phases of special functions. MATLAB Central File Exchange (2016) . http://www.mathworks.com/matlabcentral/fileexchange/57582-specialphase
  13. 13.
    Ridders, C.: A new algorithm for computing a single root of a real continuous function. IEEE Trans. Circuit. Syst. 26(11), 979–980 (1979)CrossRefzbMATHGoogle Scholar
  14. 14.
    Horsley, D.E.: Specialzeros: zeros of special functions. MATLAB Central File Exchange (2016) . http://www.mathworks.com/matlabcentral/fileexchange/57679-specialzeros
  15. 15.
    IEEE Standard for Binary Floating-Point Arithmetic: Institute of Electrical and Electronics Engineers, New York (1985). Note: Standard 754–1985Google Scholar
  16. 16.
    Wolfram Research, Mathematica: Version, 7th edn. (2008)Google Scholar
  17. 17.
    Cochran, J.A.: Remarks on the zeros of cross-product Bessel functions. J. Soc. Ind. Appl. Math. 12(3), 580–587 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gradshteĭn, I.S., Ryshik, I.: Table of Integrals, Series, and Products, 7th edn. Elsevier, Amsterdam (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Physical SciencesUniversity of TasmaniaHobartAustralia

Personalised recommendations