Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 20
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Kerimov, M. K. 2016. Studies on the zeros of Bessel functions and methods for their computation: 2. Monotonicity, convexity, concavity, and other properties. Computational Mathematics and Mathematical Physics, Vol. 56, Issue. 7, p. 1175.

    Zhao, Zhizhen Shkolnisky, Yoel and Singer, Amit 2016. Fast Steerable Principal Component Analysis. IEEE Transactions on Computational Imaging, Vol. 2, Issue. 1, p. 1.

    Privat, Yannick Trélat, Emmanuel and Zuazua, Enrique 2015. Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data. Archive for Rational Mechanics and Analysis, Vol. 216, Issue. 3, p. 921.

    Riaud, Antoine Thomas, Jean-Louis Baudoin, Michael and Bou Matar, Olivier 2015. Taming the degeneration of Bessel beams at an anisotropic-isotropic interface: Toward three-dimensional control of confined vortical waves. Physical Review E, Vol. 92, Issue. 6,

    Riaud, Antoine Thomas, Jean-Louis Charron, Eric Bussonnière, Adrien Bou Matar, Olivier and Baudoin, Michael 2015. Anisotropic Swirling Surface Acoustic Waves from Inverse Filtering for On-Chip Generation of Acoustic Vortices. Physical Review Applied, Vol. 4, Issue. 3,

    Ripoll, J.-F. Albert, J. M. and Cunningham, G. S. 2014. Electron lifetimes from narrowband wave-particle interactions within the plasmasphere. Journal of Geophysical Research: Space Physics, Vol. 119, Issue. 11, p. 8858.

    Grebenkov, D. S. and Nguyen, B.-T. 2013. Geometrical Structure of Laplacian Eigenfunctions. SIAM Review, Vol. 55, Issue. 4, p. 601.

    Nguyen, B.-T. and Grebenkov, D. S. 2013. Localization of Laplacian Eigenfunctions in Circular, Spherical, and Elliptical Domains. SIAM Journal on Applied Mathematics, Vol. 73, Issue. 2, p. 780.

    Berry, Dominic W. Hall, Michael J. W. Zwierz, Marcin and Wiseman, Howard M. 2012. Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates. Physical Review A, Vol. 86, Issue. 5,

    Liverts, Evgeny Z and Barnea, Nir 2011. Transition states and the critical parameters of central potentials. Journal of Physics A: Mathematical and Theoretical, Vol. 44, Issue. 37, p. 375303.

    Xie, Kanghe Wang, Yulin Wang, Kun and Cai, Xin 2010. Application of Hankel transforms to boundary value problems of water flow due to a circular source. Applied Mathematics and Computation, Vol. 216, Issue. 5, p. 1469.

    Muldoon, Martin E. 2008. Continuous ranking of zeros of special functions. Journal of Mathematical Analysis and Applications, Vol. 343, Issue. 1, p. 436.


    Elbert, Á. 2001. Some recent results on the zeros of Bessel functions and orthogonal polynomials. Journal of Computational and Applied Mathematics, Vol. 133, Issue. 1-2, p. 65.

    Muldoon, M. E. and Spigler, R. 1984. Some Remarks on Zeros of Cylinder Functions. SIAM Journal on Mathematical Analysis, Vol. 15, Issue. 6, p. 1231.

    Baierlein, Ralph 1976. On the electromagnetic detection of gravitational waves. General Relativity and Gravitation, Vol. 7, Issue. 7, p. 583.

    Naylor, D. and Dennis, S. C. R. 1968. On a singular eigenvalue problem. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 64, Issue. 02, p. 439.

    Oden, Lynn and Henderson, Douglas 1964. Quantum Tunnel Model for Hard Spheres. The Journal of Chemical Physics, Vol. 41, Issue. 11, p. 3487.

    Imai, I. 1956. A refinement of the WKB method and its application to the electromagnetic wave theory. IRE Transactions on Antennas and Propagation, Vol. 4, Issue. 3, p. 233.

    Olver, F. W. J. and Goodwin, E. T. 1952. Some new asymptotic expansions for Bessel functions of large orders. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 48, Issue. 03, p. 414.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 47, Issue 4
  • October 1951, pp. 699-712

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

  • F. W. J. Olver (a1)
  • DOI:
  • Published online: 24 October 2008

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder function

as a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

(6)J. R. Airey The numerical calculation of the roots of the Bessel function Jn (x) and its first derivate Phil. Mag. 34 (1917), 189–95.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *