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1: 3.11 Approximation Techniques
§3.11(i) Minimax Polynomial Approximations
Then there exists a unique n th degree polynomial p n ( x ) , called the minimax (or best uniform) polynomial approximation to f ( x ) on [ a , b ] , that minimizes max a x b | ϵ n ( x ) | , where ϵ n ( x ) = f ( x ) - p n ( x ) . …
§3.11(iii) Minimax Rational Approximations
Then the minimax (or best uniform) rational approximation
Example
2: Bibliography Q
  • W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
  • C. K. Qu and R. Wong (1999) Best possible” upper and lower bounds for the zeros of the Bessel function J ν ( x ) . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
  • 3: 14.26 Uniform Asymptotic Expansions
    §14.26 Uniform Asymptotic Expansions
    The uniform asymptotic approximations given in §14.15 for P ν - μ ( x ) and Q ν μ ( x ) for 1 < x < are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
    4: Browsers
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    5: 16.26 Approximations
    §16.26 Approximations
    For discussions of the approximation of generalized hypergeometric functions and the Meijer G -function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).
    6: Need Help?
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    7: 5.23 Approximations
    §5.23(i) Rational Approximations
    Cody et al. (1973) gives minimax rational approximations for ψ ( x ) for the ranges 0.5 x 3 and 3 x < ; precision is variable. …
    §5.23(iii) Approximations in the Complex Plane
    See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of Γ ( z ) . For rational approximations to ψ ( z ) + γ see Luke (1975, pp. 13–16).
    8: Bibliography W
  • Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
  • E. J. Weniger and J. Čížek (1990) Rational approximations for the modified Bessel function of the second kind. Comput. Phys. Comm. 59 (3), pp. 471–493.
  • H. Werner, J. Stoer, and W. Bommas (1967) Rational Chebyshev approximation. Numer. Math. 10 (4), pp. 289–306.
  • C. A. Wills, J. M. Blair, and P. L. Ragde (1982) Rational Chebyshev approximations for the Bessel functions J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) . Math. Comp. 39 (160), pp. 617–623.
  • R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.
  • 9: 4.47 Approximations
    §4.47 Approximations
    §4.47(ii) Rational Functions
    §4.47(iii) Padé Approximations
    Luke (1975, Chapter 3) supplies real and complex approximations for ln , exp , sin , cos , tan , arctan , arcsinh . …
    10: Bibliography T
  • N. M. Temme and A. B. Olde Daalhuis (1990) Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.
  • N. M. Temme (1978) Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.
  • N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.
  • N. M. Temme (1995c) Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.
  • A. Trellakis, A. T. Galick, and U. Ravaioli (1997) Rational Chebyshev approximation for the Fermi-Dirac integral F - 3 / 2 ( x ) . Solid–State Electronics 41 (5), pp. 771–773.