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    Spigler, Renato and Vianello, Marco 2012. The “phase function” method to solve second-order asymptotically polynomial differential equations. Numerische Mathematik, Vol. 121, Issue. 3, p. 565.

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

    Fabijonas, Bruce R. and Olver, F. W. J. 1999. On the Reversion of an Asymptotic Expansion and the Zeros of the Airy Functions. SIAM Review, Vol. 41, Issue. 4, p. 762.

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    Gold, Louis 1957. Inverse Bessel Functions: Solution for the Zeros. Journal of Mathematics and Physics, Vol. 36, Issue. 1-4, p. 167.

    Olver, F. W. J. and Goodwin, E. T. 1951. A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 47, Issue. 04, p. 699.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 46, Issue 4
  • October 1950, pp. 570-580

A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations

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

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their determination is the numerical integration of this differential equation. It has the advantage of being independent of tabulated values of the actual functions whose zeros are being sought. As an example of the application of the method the Bessel functions Jn(x), Yn(x) are considered. Numerical techniques for integrating the differential equation for the zeros of these Bessel functions are described in detail.

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(1)W. G. Bickley , J. C. P. Miller and C. W. Jones Notes on the evaluation of zeros and turning values of Bessel functions. Phil. Mag. 36 (1945), 121–33 and 200–10.

(5)W. G. Bickley Formulae relating to Bessel functions of moderate or large argument and order. Phil. Mag. 34 (1943), 371–49.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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