double argument

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21: Bibliography M

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The dilogarithm function for complex argument. Proc. Roy. Soc. London Ser. A 459, pp. 2807–2819.

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External Links:

ISSN 1364-5021, Document, MathReview (Wadim Zudilin), ZentralBlatt

Referenced by:

(25.12.1), (25.12.2), (25.12.3), (25.12.4), (25.12.5), (25.12.6), (25.12.7), (25.12.8), (25.12.9), §25.12(ii), §25.12(i), §25.12(i), §25.12(i), §25.12.

Encodings:

BibTeX

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

(25.11.21), (25.11.22), (25.11.23), §25.11, §25.11(vi), §25.18(i), (25.6.15).

Encodings:

BibTeX

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The dilogarithm function of a real argument. Math. Comp. 33 (146), pp. 778–787.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item, 4th item.

Encodings:

BibTeX

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Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Modelling 57 (5-6), pp. 1360–1363.

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External Links:

ISSN 0895-7177, Document, FullText, MathReview (Emil C. Popa)

Referenced by:

§5.6(i).

Encodings:

BibTeX

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22: 25.6 Integer Arguments

§25.6 Integer Arguments

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§25.6(i) Function Values

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25.6.7 $$\zeta \left(2\right)={\int }_0^1{\int }_0^1\frac{1}{1-xy}dxdy.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Keywords:

double integral

Source:

Apostol (1983, p. 59)

Permalink:

http://dlmf.nist.gov/25.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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§25.6(ii) Derivative Values

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23: Bibliography H

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Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§35.1, §35.2, §35.3(ii), §35.5(i), §35.5(ii), §35.5(iii), §35.6(i), §35.6(ii), §35.6(iii), §35.6(iv), §35.7(i), §35.7(ii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), Ch.35, Notations P.

Encodings:

BibTeX

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Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.

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Notes:

Double-precision Fortran, maximum accuracy 12D.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

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Two-center Coulomb functions. Comput. Phys. Comm. 103 (2-3), pp. 209–216.

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Notes:

Double-precision Fortran code.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

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24: 3.5 Quadrature

… ►A second example is provided in Gil et al. (2001), where the method of contour integration is used to evaluate Scorer functions of complex argument (§9.12). … ►

3.5.47 $$\frac{1}{\pi h^2}{\iint }_{D}f(x,y)dxdy=\sum _{j=1}^n{w}_{j}f(x_j,y_j)+R,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $(a,b)$: open interval, $D$: disk, $R$: remainder and $w_k$: weights

A&S Ref:

25.4.61

Permalink:

http://dlmf.nist.gov/3.5.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(x), §3.5 and Ch.3

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3.5.48 $$\frac{1}{4h^2}{\iint }_{S}f(x,y)dxdy=\sum _{j=1}^n{w}_{j}f(x_j,y_j)+R.$$

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Symbols:

$dx$: differential of $x$ , $(a,b)$: open interval, $R$: remainder and $w_k$: weights

A&S Ref:

25.4.62

Permalink:

http://dlmf.nist.gov/3.5.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(x), §3.5 and Ch.3

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25: 2.11 Remainder Terms; Stokes Phenomenon

… ►However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►For large $|z|$, with $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$ (), the Whittaker function of the second kind has the asymptotic expansion (§13.19) … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

26: 34.4 Definition: $\mathit{6}j$ Symbol

… ►The $\mathit{6}j$ symbol is defined by the following double sum of products of $\mathit{3}j$ symbols: … ►where the summation is over all nonnegative integers $s$ such that the arguments in the factorials are nonnegative. … ►For alternative expressions for the $\mathit{6}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_4{}^{}F_3^{}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).