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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/, Release 1.0.13 of 2016-09-09, online companion to [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]
McMahon, J.: On the roots of the Bessel and certain related functions. Ann. Math. 9(1), 23–30 (1894)
Segura, J.: A global Newton method for the zeros of cylinder functions. Numer. Algorithm 18(3), 259–276 (1998). doi:10.1023/A:1019125616736
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
Segura, J.: Computing the complex zeros of special functions. Numer. Math. 124(4), 723–752 (2013). doi:10.1007/s00211-013-0528-6
Watson, G.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (1966)
Hankel, H.: Cylinderfunktionen erster und zweiter Art. Math. Ann. 1(3), 467–501 (1869)
Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)
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)
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
Horsley, D.E.: Specialphase: phases of special functions. MATLAB Central File Exchange (2016) . http://www.mathworks.com/matlabcentral/fileexchange/57582-specialphase
Ridders, C.: A new algorithm for computing a single root of a real continuous function. IEEE Trans. Circuit. Syst. 26(11), 979–980 (1979)
Horsley, D.E.: Specialzeros: zeros of special functions. MATLAB Central File Exchange (2016) . http://www.mathworks.com/matlabcentral/fileexchange/57679-specialzeros
IEEE Standard for Binary Floating-Point Arithmetic: Institute of Electrical and Electronics Engineers, New York (1985). Note: Standard 754–1985
Wolfram Research, Mathematica: Version, 7th edn. (2008)
Cochran, J.A.: Remarks on the zeros of cross-product Bessel functions. J. Soc. Ind. Appl. Math. 12(3), 580–587 (1964)
Gradshteĭn, I.S., Ryshik, I.: Table of Integrals, Series, and Products, 7th edn. Elsevier, Amsterdam (2007)
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.
