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Some new asymptotic expansions for Bessel functions of large orders

  • F. W. J. Olver (a1)

During the course of recent work (6) on the zeros of the Bessel functions Jn(x) and Yn(x), it became evident that the theory of the asymptotic expansion of Bessel functions whose arguments and orders are of comparable magnitudes was incomplete. The existing expansions for large orders are those of Debye and Meissel, detailed derivations of both of which are given by Watson ((8), pp. 237–48).

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(1)British Association Mathematical Tables, part-vol. B, The Airy integral (Cambridge, 1946).
(2)Copson, E. T.Theory of functions (Oxford, 1935), pp. 323–4.
(3)Imai, I.Asymptotic solutions of ordinary differential equations of the second order. Phys. Rev. (2), 80 (1950), 1112.
(4)Langer, R. E.On the asymptotic solutions of ordinary differential equations with application to Bessel functions of large order. Trans. Amer. math. Soc. 33 (1931), 2364.
(5)Nicholson, J. W.The asymptotic expansion of Bessel functions. Phil. Mag. (6), 19 (1910), 228–49.
(6)Olver, F. W. J.A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Camb. phil. Soc. 47 (1951), 699712.
(7)Watson, G. N.Bessel functions of large order. Proc. Camb. phil. Soc. 19 (1918), 96110.
(8)Watson, G. N.Theory of Bessel functions (Cambridge, 1944).
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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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