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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Miniaturized tables of Bessel functions
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by Yudell L. Luke PDF
Math. Comp. 25 (1971), 323-330 Request permission


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.
    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. Ibid., Vol. 2, p. 26. Ibid., Vol. 1, pp. 308-314. Ibid., Vol. 2, pp. 339, 341, 360, 362, 365, 367.
  • 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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 323-330
  • MSC: Primary 65A05
  • DOI:
  • MathSciNet review: 0295508