asymptotic approximations of sums and sequences

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1: 2.10 Sums and Sequences

§2.10 Sums and Sequences

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§2.10(iii) Asymptotic Expansions of Entire Functions

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§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

… ►See also Flajolet and Odlyzko (1990).

2: 2.1 Definitions and Elementary Properties

… ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. … ►

§2.1(iv) Uniform Asymptotic Expansions

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§2.1(v) Generalized Asymptotic Expansions

… ►Then $\{{\varphi }_s(x)\}$ is an asymptotic sequence or scale. Suppose also that $f(x)$ and $f_s(x)$ satisfy …

3: 5.4 Special Values and Extrema

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5.4.7 $$\mathrm{\Gamma }\left(\frac{1}{3}\right)=\mathrm{2.67893\hspace{0.33em}85347\hspace{0.33em}07747\hspace{0.33em}63365}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.11 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073005; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.8 $$\mathrm{\Gamma }\left(\frac{2}{3}\right)=\mathrm{1.35411\hspace{0.33em}79394\hspace{0.33em}26400\hspace{0.33em}41694}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.13 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073006; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.9 $$\mathrm{\Gamma }\left(\frac{1}{4}\right)=\mathrm{3.62560\hspace{0.33em}99082\hspace{0.33em}21908\hspace{0.33em}31193}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.10 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A068466; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

… ►As $n\to \mathrm{\infty }$, ►

5.4.20 $$x_n=-n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n}\right)+O\left(\frac{1}{n{(\ln n)}^2}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

6.3.20 (The error estimate has been improved.)

Referenced by:

§5.4(iii)

Permalink:

http://dlmf.nist.gov/5.4.E20

Encodings:

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See also:

Annotations for §5.4(iii), §5.4 and Ch.5

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4: Bibliography W

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Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

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Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

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External Links:

ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

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

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

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Error bounds for asymptotic approximations of special functions. Ann. Numer. Math. 2 (1-4), pp. 181–197.

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Notes:

Special functions (Torino, 1993)

External Links:

ISSN 1021-2655, MathReview (Andrei Martínez Finkelshtein), ZentralBlatt

Referenced by:

§10.21(vi).

Encodings:

BibTeX

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5: 4.45 Methods of Computation

… ►Beginning with $x_0=x$, generate the sequence … ►Another method, when $x$ is large, is to sum … ►Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

6: Bibliography C

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Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.

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External Links:

ISSN 0036-1410, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.12, §19.27(ii), §19.27(iii), §19.27(iv), §19.27(v), §19.27(vi).

Encodings:

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Introduction to Approximation Theory. 2nd edition, Chelsea Publishing Co., New York.

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Notes:

Reprinted by AMS Chelsea Publishing, Providence, RI, 1998

External Links:

ISBN 0-8218-1374-9, MathReview, ZentralBlatt

Referenced by:

§18.38(i).

Encodings:

BibTeX

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Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

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Notes:

Double-precision Fortran, maximum accuracy 24S.

External Links:

ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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Department of Computer Science, University of Victoria, Canada.

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Notes:

Permutations, set and integer partitions, sequences, graphs, and trees.

External Links:

Combinatorial Object Server

Referenced by:

1st item.

Encodings:

BibTeX

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A sequence of series for the Lambert $W$ function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.

Encodings:

BibTeX

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7: Bibliography S

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Calculator function approximation. Amer. Math. Monthly 90 (5), pp. 317–325.

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External Links:

ISSN 0002-9890, FullText, MathReview (J. Albrycht), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

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The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

BibTeX

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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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The On-Line Encyclopedia of Integer Sequences. Notices Amer. Math. Soc. 50 (8), pp. 912–915.

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External Links:

ISSN 0002-9920, FullText, Electronic Database, MathReview (Christian Krattenthaler), ZentralBlatt

Referenced by:

(19.20.2), (19.20.22), (19.30.13), (25.11.40), (3.12.1), (3.12.3), (3.12.4), §3.12, (4.2.11), (4.2.17), (4.2.18), (4.4.19), (5.17.6), (5.2.3), (5.4.10), (5.4.6), (5.4.7), (5.4.8), (5.4.9), (6.13.1).

Encodings:

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Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:

ISBN 0130438936;978-0130438935, MathReview (R. D. Riess), ZentralBlatt

Referenced by:

§3.5(x).

Encodings:

BibTeX

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8: 18.40 Methods of Computation

… ►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. … ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to $w(x)$, as will be considered in the following paragraphs. … ►

Stieltjes Inversion via (approximate) Analytic Continuation

… ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. … ►In Figure 18.40.2 the approximations were carried out with a precision of 50 decimal digits.

9: Bibliography K

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Unimodal sequences show that Lambert $W$ is Bernstein. C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.

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External Links:

ISSN 0706-1994, MathReview (Fana Tangara), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

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

Encodings:

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

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External Links:

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

Referenced by:

§18.24, §18.24.

Encodings:

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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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External Links:

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

Referenced by:

§2.4(vi), §2.4(vi).

Encodings:

BibTeX

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

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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10: Bibliography L

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Asymptotic Analysis. Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam.

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External Links:

MathReview (C. Comstock), ZentralBlatt

Referenced by:

Ch.2.

Encodings:

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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

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Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

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Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

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Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

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External Links:

MathReview (F. Gotze), ZentralBlatt

Referenced by:

§13.31(iii).

Encodings:

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