best uniform rational approximation

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21—30 of 235 matching pages

21: Bibliography J

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M. Jimbo and T. Miwa (1981) Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2 (3), pp. 407–448.

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External Links:: ISSN 0167-2789, Document, MathReview (V. A. Golubeva), ZentralBlatt
Referenced by:: §32.4(i), §32.4(ii), §32.4(iv), §32.4(v), §32.6(ii), §32.6(iii), §32.6(iv), §32.6(v), §32.6(v), §32.6(v).
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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External Links:: ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §18.24, §2.4(vi).
Encodings:: BibTeX

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J. H. Johnson and J. M. Blair (1973) REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.

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Referenced by:: §3.11(iii).
Encodings:: BibTeX

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S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions

\overline{ps}\,^{r}_{n}(\eta,h)

and

\overline{qs}\,^{r}_{n}(\eta,h)

for large

h

. Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §30.9(ii).
Encodings:: BibTeX

…

22: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

… ►

§28.26(ii) Uniform Approximations

… ►For asymptotic approximations for

{\operatorname{M}^{(3,4)}_{\nu}}\left(z,h\right)

23: 9.16 Physical Applications

… ►The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. …The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). … ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. … ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. … ►An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).

24: DLMF Project News

error generating summary

25: Bibliography P

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B. V. Pal^′tsev (1999) 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.
Encodings:: BibTeX

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R. B. Paris (2002a) 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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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

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R. B. Paris (2002b) A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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External Links:: 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).
Encodings:: BibTeX

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S. Paszkowski (1988) Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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R. Piessens (1984a) Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

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External Links:: ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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26: Bibliography O

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A. B. Olde Daalhuis and N. M. Temme (1994) Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

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External Links:: ISSN 0036-1410, Document, MathReview, ZentralBlatt
Referenced by:: §2.4(v).
Encodings:: BibTeX

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A. B. Olde Daalhuis (1998c) 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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External Links:: ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

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A. B. Olde Daalhuis (2000) 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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External Links:: ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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F. W. J. Olver (1980b) 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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External Links:: ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt
Referenced by:: §13.20(iii), §13.20(iv).
Encodings:: BibTeX

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27: Bibliography S

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T. Schmelzer and L. N. Trefethen (2007) Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45 (2), pp. 558–571.

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External Links:: ISSN 0036-1429, Document, MathReview (Amparo Gil), ZentralBlatt
Referenced by:: §5.21, §5.23(iii).
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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A. Sharples (1971) Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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External Links:: MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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H. Skovgaard (1966) Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions. Doctoral dissertation, University of Copenhagen, Vol. 1965, Jul. Gjellerups Forlag, Copenhagen.

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External Links:: MathReview (A. K. Agarwal), ZentralBlatt
Referenced by:: §13.21(iv).
Encodings:: BibTeX

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W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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External Links:: Abstract
Referenced by:: §18.16(v).
Encodings:: BibTeX

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28: 33.25 Approximations

§33.25 Approximations

►Cody and Hillstrom (1970) provides rational approximations of the phase shift

{\sigma_{0}}\left(\eta\right)=\operatorname{ph}\Gamma\left(1+\mathrm{i}\eta\right)

(see (33.2.10)) for the ranges

0\leq\eta\leq 2

2\leq\eta\leq 4

, and

4\leq\eta\leq\infty

. …

29: 18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

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18.29.2

Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},

n\to\infty

;

z, a, b, c, d, q

fixed.

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Symbols:: $\sim$ : asymptotic equality, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $z$ : complex variable, $q$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.29.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.29 and Ch.18

►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the

q

-Laguerre and continuous

q^{-1}

-Hermite polynomials see Chen and Ismail (1998).

30: 28.8 Asymptotic Expansions for Large $q$

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§28.8(iv) Uniform Approximations

►

Barrett’s Expansions

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Dunster’s Approximations

►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►