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Mathematics of Computation

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Miniaturized tables of Bessel functions

Author: Yudell L. Luke
Journal: Math. Comp. 25 (1971), 323-330
MSC: Primary 65A05
MathSciNet review: 0295508
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Abstract: In this report, we discuss the representation of bivariate functions in double series of Chebyshev polynomials. For an application, we tabulate coefficients which are accurate to 20 decimals for the evaluation of $ {(2z/\pi )^{1/2}}{e^z}{K_v}(z)$ for all $ z \geqq 5$ and all $ v,0 \leqq v \leqq 1$. Since $ {K_v}(z)$ is an even function in v and satisfies a three-term recurrence formula in v which is stable when used in the forward direction, we can readily evaluate $ {K_v}(z)$ for all $ z \geqq 5$ and all $ v \geqq 0$. Only 205 coefficients are required to achieve an accuracy of about 20 decimals for the z and v ranges described. Extension of these ideas for the evaluation of all Bessel functions and other important bivariate functions is under way.

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  • [1] Y. L. Luke, The Special Functions and Their Approximations. Vols. 1, 2, Mathematics in Science and Engineering, vol. 53, Academic Press, New York, 1969. See especially Vol. 1, p. 213; Vol. 2, pp. 25-28. MR 39 #3039; MR 40 #2909.
  • [2] Ibid., Vol. 2, p. 26.
  • [3] Ibid., Vol. 1, pp. 308-314.
  • [4] Ibid., Vol. 2, pp. 339, 341, 360, 362, 365, 367.
  • [5] C. W. Clenshaw and Susan M. Picken, Chebyshev series for Bessel functions of fractional order, National Physical Laboratory Mathematical Tables, Vol. 8, Her Majesty’s Stationery Office, London, 1966. MR 0203095

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Keywords: Bessel functions, approximation of bivariate functions, approximation in series of Chebyshev polynomials
Article copyright: © Copyright 1971 American Mathematical Society