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1: 3.11 Approximation Techniques

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§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 ${\max }_{a\le x\le b}\left|{ϵ}_n(x)\right|$, where ${ϵ}_n(x)=f(x)-p_n(x)$. … ►

§3.11(iii) Minimax Rational Approximations

… ►Then the minimax (or best uniform) rational approximation … ►

Example

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2: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

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3: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for 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

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§5.23(i) Rational Approximations

… ►Cody et al. (1973) gives minimax rational approximations for $\psi \left(x\right)$ for the ranges $0.5\le x\le 3$ and ; 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 $\mathrm{\Gamma }\left(z\right)$. ►For rational approximations to $\psi \left(z\right)+\gamma$ see Luke (1975, pp. 13–16).

8: Bibliography W

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

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§18.29.

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Rational approximations for the modified Bessel function of the second kind. Comput. Phys. Comm. 59 (3), pp. 471–493.

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ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

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Rational Chebyshev approximation. Numer. Math. 10 (4), pp. 289–306.

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ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.11(iii).

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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.

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Notes:

Tables on microfiche

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

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9th item.

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

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9: 4.47 Approximations

§4.47 Approximations

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§4.47(ii) Rational Functions

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§4.47(iii) Padé Approximations

►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …

10: Bibliography T

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Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii).

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Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

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ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

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Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.

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ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

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Ch.2.

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Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.

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Document

Referenced by:

§25.21(vii).

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