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1: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

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

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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Coulomb Functions for Large Values of the Parameter $\eta$ . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.

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Notes:

Rep. CRT-526

External Links:

MathReview (A. Erdélyi)

Referenced by:

§33.7.

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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2: 18.40 Methods of Computation

… ►There are many ways to implement these first two steps, noting that the expressions for ${\alpha }_n$ and ${\beta }_n$ of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010). … ►It is now necessary to take the limit $\epsilon \to 0+$ of $F(x^{\prime }+\mathrm{i}\epsilon )$, and the imaginary part is the required Stieltjes–Perron inversion: ►

18.40.6 $$\underset{\epsilon \to 0+}{lim}{\int }_a^b\frac{w(x)dx}{x^{\prime }+\mathrm{i}\epsilon -x}dx={⨍}_a^b\frac{w(x)dx}{x^{\prime }-x}-\mathrm{i}\pi w(x^{\prime }),$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${⨍}_a^b$: Cauchy principal value, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. Gautschi (2004, p. 119–120) has explored the $\epsilon \to 0^{+}$ limit via the Wynn $\epsilon$-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in $w(x)$, depending smoothly on $x^{\prime }$, for $N\approx 4000$, for an example involving first numerator Legendre OP’s. …

3: 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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3	6	20	75278	37	903 79784
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§26.12(iv) Limiting Form

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4: 6.16 Mathematical Applications

… ►These limits are not approached uniformly, however. …Hence if $x=\pi /(2n)$ and $n\to \mathrm{\infty }$, then the limiting value of $S_n(x)$ overshoots $\frac{1}{4}\pi$ by approximately 18%. Similarly if $x=\pi /n$, then the limiting value of $S_n(x)$ undershoots $\frac{1}{4}\pi$ by approximately 10%, and so on. … ►

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

5: 19.36 Methods of Computation

… ►Complex values of the variables are allowed, with some restrictions in the case of $R_J$ that are sufficient but not always necessary. … ►Accurate values of $F(\varphi ,k)-E(\varphi ,k)$ for $k^2$ near 0 can be obtained from $R_D$ by (19.2.6) and (19.25.13). … ►As $n\to \mathrm{\infty }$, $c_n$, $a_n$, and $t_n$ converge quadratically to limits $0$, $M$, and $T$, respectively; hence … ►This method loses significant figures in $\rho$ if ${\alpha }^2$ and $k^2$ are nearly equal unless they are given exact values—as they can be for tables. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

6: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:

ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

Encodings:

BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

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7: 26.10 Integer Partitions: Other Restrictions

… ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. … ►

§26.10(v) Limiting Form

… ►The quantity $A_k(n)$ is real-valued. …