Rational Chebyshev approximations for the Bessel functions $J_{0}(x)$, $J_{1}(x)$, $Y_{0}(x)$, $Y_{1}(x)$
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 by C. A. Wills, J. M. Blair and P. L. Ragde PDF
 Math. Comp. 39 (1982), 617623 Request permission
Abstract:
This report presents nearminimax rational approximations for the Bessel functions ${J_0}(x)$, ${J_1}(x)$, ${Y_0}(x)$, and ${Y_1}(x)$ for the complete range of x, with relative errors ranging down to ${10^{  23}}$. The first thirty zeros of each function are listed to 35D. The tabulated zeros and the McMahon asymptotic formulae may be used to construct an algorithm which retains relative accuracy in the neighborhood of zeros.References

M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, Nat. Bur. Standards, Appl. Math. Series No. 55, U. S. Government Printing Office, Washington, D. C., 1965.
R. P. Brent, "A FORTRAN MultiplePrecision Arithmetic Package," ACM Trans. Math. Software, v. 4, 1978, pp. 5770.
 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 W. J. Cody, "The FUNPACK Package of Special Function Subroutines," ACM Trans. Math. Software, v. 1, 1975, pp. 1325. W. J. Cody, R. M. Motley & L. W. Fullerton, "Coefficients for the approximation of ${Y_v}(x)$," AMD Technical Memorandum #284, Argonne National Laboratory. (In preparation.)
 Boro Döring, Über die McMahonEntwicklungen, Z. Angew. Math. Phys. 18 (1967), 461–473 (German, with English summary). MR 214272, DOI 10.1007/BF01601717 J. F. Hart, et al., Computer Approximations, Wiley, New York, 1968. K. Hayashi, Tafeln der Besselschen, Theta, Kugel—und anderer Funktionen, SpringerVerlag, Berlin, 1930. J. H. Johnson & J. M. Blair, REMES2: A FORTRAN Program to Calculate Rational Minimax Approximations to a Given Function, Atomic Energy of Canada Limited, Report AECL4210, Chalk River, Ontario, 1973.
 Yudell L. Luke, Mathematical functions and their approximations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1975. MR 0501762 S. Makinouchi, Zeros of Bessel Functions ${J_v}(x)$ and ${Y_v}(x)$ Accurate to TwentyNine Significant Digits, Osaka University Technology Report No. 685, 1965.
 C. Mesztenyi and C. Witzgall, Stable evaluation of polynomials, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 11–17. MR 212994, DOI 10.6028/jres.071B.003
 Bessel functions. Part III: Zeros and associated values, Royal Society Mathematical Tables, Vol. 7, Cambridge University Press, New York, 1960. Prepared under the direction of the Bessel Functions Panel of the Mathematical Tables Committee. MR 0119441
 Jet Wimp, Polynomial expansions of Bessel functions and some associated functions, Math. Comp. 16 (1962), 446–458. MR 148956, DOI 10.1090/S00255718196201489563
Additional Information
 © Copyright 1982 American Mathematical Society
 Journal: Math. Comp. 39 (1982), 617623
 MSC: Primary 65D20; Secondary 33A40, 41A50
 DOI: https://doi.org/10.1090/S00255718198206696534
 MathSciNet review: 669653