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31: 28.26 Asymptotic Approximations for Large $q$

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28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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32: 26.6 Other Lattice Path Numbers

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Table 26.6.4: Schröder numbers $r(n)$.

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$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
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$3$	$22$	$7$	$8558$	$11$	$\mathrm{52\hspace{0.33em}93446}$	$15$	$\mathrm{39376\hspace{0.33em}03038}$	$19$	$\mathrm{323\hspace{0.33em}67243\hspace{0.33em}17174}$

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33: Bibliography V

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A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.

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External Links:

MathReview (J. V. Wehausen), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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A new estimate of the function $\zeta (1+it)$ . Izv. Akad. Nauk SSSR. Ser. Mat. 22, pp. 161–164 (Russian).

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External Links:

MathReview (H. Davenport), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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34: 19.12 Asymptotic Approximations

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35: 20.9 Relations to Other Functions

… ►See Koblitz (1993, Ch. 2, §4) and Titchmarsh (1986b, pp. 21–22). …

36: 2.1 Definitions and Elementary Properties

… ►Substitution, logarithms, and powers are also permissible; compare Olver (1997b, pp. 19–22). …

37: 3.7 Ordinary Differential Equations

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3.7.6

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Defines:

$𝐀(\tau ,z)$: matrix (locally) and $A_{jk}(\tau ,z)$: matrix entry (locally)

Permalink:

http://dlmf.nist.gov/3.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(ii), §3.7 and Ch.3

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$$A_{22}(\tau ,z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\tau }^s}{s!}{g}_{s+1}(z),$$

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38: 26.12 Plane Partitions

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Table 26.12.1: Plane partitions.

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$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$
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5	24	22	1 86475	39	1934 87501
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39: Bibliography L

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Higher monotonicity properties of certain Sturm-Liouville functions. III. Canad. J. Math. 22, pp. 1238–1265.

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External Links:

ISSN 0008-414X, MathReview (G. Sansone), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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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:

BibTeX

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40: Bibliography W

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Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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External Links:

FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt

Referenced by:

§16.3(ii).

Encodings:

BibTeX

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Concerning two series for the gamma function. Math. Comp. 22 (103), pp. 617–626.

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External Links:

FullText, Document, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§5.11(i), §5.7(i), §5.7(i).

Encodings:

BibTeX

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