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11—20 of 93 matching pages

11: Staff

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Karl Dilcher, Dalhousie University, Chap. 24

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Karl Dilcher, Dalhousie University, for Chap. 24

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12: 23.19 Interrelations

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23.19.4 $$\mathrm{\Delta }={(2\pi )}^{12}{\eta }^{24}\left(\tau \right).$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Delta }$: discriminant and $\tau$: complex variable

Referenced by:

§23.19

Permalink:

http://dlmf.nist.gov/23.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

13: 25.20 Approximations

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Morris (1979) gives rational approximations for ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $0.5\le x\le 1$. Precision is varied with a maximum of 24S.

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14: 27.16 Cryptography

… ►For further information see Apostol and Niven (1994, p. 24), and for other applications to cryptography see Menezes et al. (1997) and Schroeder (2006).

15: Richard A. Askey

… ►Askey was presented a Lifetime Achievement Award in Recognition and Appreciation for his Outstanding Work and Leadership in the Field of Special Functions at the International Symposium on Orthogonal Polynomials, Special Functions and Applications in Hagenberg, Austria on July 24, 2019. …

16: Bibliography K

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

BibTeX

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Calculation of modified Bessel functions in a complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(3), pp. 15–24

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(iv).

Encodings:

BibTeX

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Calculation of the complex zeros of the modified Bessel function of the second kind and its derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 24 (8), pp. 1150–1163.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(4), pp. 115–123

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(vi), 3rd item.

Encodings:

BibTeX

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Evaluation of complex zeros of Bessel functions $J_{\nu }(z)$ and $I_{\nu }(z)$ and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 24 (10), pp. 1497–1513.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(5), pp. 131–141

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 4th item.

Encodings:

BibTeX

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

BibTeX

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17: 3.9 Acceleration of Convergence

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Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

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$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
0	0.80000 00000 00	0.82182 62806 24	0.82244 84501 47	0.82246 64909 60	0.82246 70175 41
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5	0.82259 80392 16	0.82246 88857 22	0.82246 70670 21	0.82246 70341 24	0.82246 70334 40
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9	0.82248 70624 89	0.82246 71865 91	0.82246 70351 34	0.82246 70334 48	0.82246 70334 24
10	0.82245 30535 15	0.82246 69397 57	0.82246 70324 88	0.82246 70334 12	0.82246 70334 24

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… ►For examples and other transformations for convergent sequences and series, see Wimp (1981, pp. 156–199), Brezinski and Redivo Zaglia (1991, pp. 55–72), and Sidi (2003, Chapters 6, 12–13, 15–16, 19–24, and pp. 483–492). …

18: 27.13 Functions

… ►By similar methods Jacobi proved that $r_4\left(n\right)=8{\sigma }_1\left(n\right)$ if $n$ is odd, whereas, if $n$ is even, $r_4\left(n\right)=24$ times the sum of the odd divisors of $n$. …Exact formulas for $r_k\left(n\right)$ have also been found for $k=3,5$, and $7$, and for all even $k\le 24$. For values of $k>24$ the analysis of $r_k\left(n\right)$ is considerably more complicated (see Hardy (1940)). …

19: 26.2 Basic Definitions

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Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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7	15	24	1575	41	44583
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20: 27.7 Lambert Series as Generating Functions

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