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11: 21.1 Special Notation

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$g,h$	positive integers.
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$\mathrm{diag}𝐀$	Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$.
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12: 26.9 Integer Partitions: Restricted Number and Part Size

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Table 26.9.1: Partitions $p_k\left(n\right)$.

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$n$	$k$
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8	0	1	5	10	15	18	20	21	22	22	22
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13: 7.24 Approximations

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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14: 34.1 Special Notation

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15: Frank W. J. Olver

… ►He continued his editing work until the time of his death on April 22, 2013 at age 88.

16: Bibliography O

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Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.

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External Links:

ISSN 0036-1445, Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

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Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22 (5), pp. 1460–1474.

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External Links:

ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§2.11(iii), §2.11(iii), §8.11(i), §8.20(i).

Encodings:

BibTeX

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Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22 (5), pp. 1475–1489.

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External Links:

ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

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Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

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Notes:

Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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17: 21.5 Modular Transformations

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21.5.7

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $g$: positive integer, $𝛀$: a Riemann matrix, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Referenced by:

(21.5.8), Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/21.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

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21.5.9 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝐃𝜶-𝐂𝜷+\frac{1}{2}\mathrm{diag}[𝐂{𝐃}^{\mathrm{T}}]}{-𝐁𝜶+𝐀𝜷+\frac{1}{2}\mathrm{diag}[𝐀{𝐁}^{\mathrm{T}}]}\right]\left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\kappa (𝜶,𝜷,𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(ii), §21.5 and Ch.21

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18: 7.23 Tables

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Finn and Mugglestone (1965) includes the Voigt function $H(a,u)$, $u\in [0,22]$, $a\in [0,1]$, 6S.

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19: 10.12 Generating Function and Associated Series

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20: 23.17 Elementary Properties

… ►See also Apostol (1990, p. 22). …