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31: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

… ►However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). …

32: Bibliography P

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On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).

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Notes:

English translation in Math. Notes 65 (1999) pp. 571-581.

External Links:

ISSN 0025-567X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.37.

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Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

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Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

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ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.76(ii).

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

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33: Bibliography O

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Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

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ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

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On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.

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ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§8.12.

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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

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34: 18.35 Pollaczek Polynomials

§18.35 Pollaczek Polynomials

… ►There are 3 types of Pollaczek polynomials: … ►For the monic polynomials … ►This expansion is in terms of the Airy function $\mathrm{Ai}\left(x\right)$ and its derivative (§9.2), and is uniform in any compact $\theta$-interval in $(0,\mathrm{\infty })$. Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$. …

35: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

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Mathematical Approximation of Special Functions. Nova Science Publishers Inc., Commack, NY.

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Notes:

Ten papers on Chebyshev expansions

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ISBN 1-56072-052-2, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.76(ii), §10.76(ii), §10.76(ii), §10.76(iii), 2nd item.

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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§2.8(vi).

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Approximations for the Bessel and Struve functions. Math. Comp. 43 (168), pp. 551–556.

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ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

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1st item.

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Askey-Wilson polynomials: an affine Hecke algebra approach. In Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 111–144.

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MathReview (V. L. Deshpande), ZentralBlatt

Referenced by:

§18.38(iii).

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36: Bibliography K

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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. J. Comput. Appl. Math. 205 (1), pp. 186–206.

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ISSN 0377-0427, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.12, §19.36(iv), §19.36(iv).

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Uniform asymptotic approximations for the Meixner-Sobolev polynomials. Anal. Appl. (Singap.) 10 (3), pp. 345–361.

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ISSN 0219-5305, Document, FullText, MathReview (Diego E. Dominici), ZentralBlatt

Referenced by:

§18.24, §18.24.

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Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2153), pp. 20130008, 12.

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ISSN 1364-5021, Document, FullText, MathReview (Ulrich D. Jentschura), ZentralBlatt

Referenced by:

§2.4(vi), §2.4(vi).

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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

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MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

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MathReview, ZentralBlatt

Referenced by:

§3.5(viii).

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37: 19.38 Approximations

§19.38 Approximations

►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

38: 18.40 Methods of Computation

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§18.40(i) Computation of Polynomials

►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. … ►

Stieltjes Inversion via (approximate) Analytic Continuation

… ►In Figure 18.40.2 the approximations were carried out with a precision of 50 decimal digits.

39: 7.24 Approximations

§7.24 Approximations

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§7.24(i) Approximations in Terms of Elementary Functions

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Hastings (1955) gives several minimax polynomial and rational approximations for $\mathrm{erf}x$, $\mathrm{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$.

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Cody (1969) provides minimax rational approximations for $\mathrm{erf}x$ and $\mathrm{erfc}x$. The maximum relative precision is about 20S.

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Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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40: 36.15 Methods of Computation

… ►Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …