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11: Bibliography U

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Mathematics of Computation Unpublished Mathematical Tables Collection.

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Notes:

Archives of American Mathematics, Center for American History, The University of Texas at Austin. Covers 1944–1994. An online guide is arranged chronologically by date of issuance within Mathematics of Computation

External Links:

Guide

Referenced by:

§27.21.

Encodings:

BibTeX

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On Kelvin’s ship-wave pattern. J. Fluid Mech. 8 (3), pp. 418–431.

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External Links:

Document, MathReview (R. C. MacCamy), ZentralBlatt

Referenced by:

§36.13.

Encodings:

BibTeX

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Theory of the Whittaker-Hill equation. Proc. Roy. Soc. Edinburgh Sect. A 69, pp. 28–44.

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External Links:

MathReview (R. K. Rathy), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

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12: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►Table 26.4.1 gives numerical values of multinomials and partitions $\lambda ,M_1,M_2,M_3$ for $1\le m\le n\le 5$. …$M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: ►

26.4.8 $$M_3=\frac{n!}{{(1!)}^{a_1}(a_1!){(2!)}^{a_2}(a_2!)\mathrm{\cdots }{(n!)}^{a_n}(a_n!)}.$$

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Symbols:

$!$: factorial (as in $n!$), $n$: nonnegative integer, $a_n$: nonnegative integers and $M_3$: number of partitions

Permalink:

http://dlmf.nist.gov/26.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(i), §26.4 and Ch.26

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Table 26.4.1: Multinomials and partitions.

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$n$	$m$	$\lambda$	$M_1$	$M_2$	$M_3$
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$3$	$2$	$1^1,2^1$	$3$	$3$	$3$
$3$	$3$	$1^3$	$6$	$1$	$1$
…

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13: 26.2 Basic Definitions

… ►Thus $231$ is the permutation $\sigma (1)=2$, $\sigma (2)=3$, $\sigma (3)=1$. … ►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$. …The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. … ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►

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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11	56	28	3718	45	89134
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14: 25.14 Lerch’s Transcendent

… ►With the conditions of (25.14.1) and $m=1,2,3,\mathrm{\dots }$, … ►

25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

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Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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15: 27.2 Functions

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Table 27.2.1: Primes.

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$n$	$p_n$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
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2	3	37	79	131	181	239	293	359	421	479
3	5	41	83	137	191	241	307	367	431	487
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Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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2	1	2	3	15	8	4	24	28	12	6	56	41	40	2	42
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
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12	4	6	28	25	20	3	31	38	18	4	60	51	32	4	72
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16: Bibliography P

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Notes:

Computer algebra system designed for computations in number theory. Many transcendental functions are also included. PARI is also available as a C library to allow for faster computations. Originally developed by Henri Cohen and co-workers at the Université Bordeaux I.

External Links:

Home Page

Referenced by:

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

Encodings:

BibTeX

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Stokes phenomenon demystified. Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.

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External Links:

ISSN 0905-5628, MathReview Entry, ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

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External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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A Fortran routine for the computation of gamma percentiles. Adv. Eng. Software 10 (3), pp. 159–164.

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External Links:

Document, ZentralBlatt

Referenced by:

§8.28(ii).

Encodings:

BibTeX

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Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

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External Links:

ISSN 0377-0427, Document, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

BibTeX

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17: 26.6 Other Lattice Path Numbers

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Table 26.6.2: Motzkin numbers $M(n)$.

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$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
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3	4	7	127	11	5798	15	3 10572	19	181 99284

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… ► $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … ►

Table 26.6.3: Narayana numbers $N(n,k)$.

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$n$	$k$
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3	0	1	3	1
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8	0	1	28	196	490	490	196	28	1
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26.6.7 $$\sum _{n=0}^{\mathrm{\infty }}M(n)x^n=\frac{1-x-\sqrt{1-2x-3x^2}}{2x^2},$$

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Symbols:

$n$: nonnegative integer, $M(n)$: Motzkin number and $x$: small

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(ii), §26.6 and Ch.26

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18: 23.9 Laurent and Other Power Series

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$c_3={\displaystyle \frac{1}{28}}{g}_3,$

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23.9.5 $$c_n=\frac{3}{(2n+1)(n-3)}\sum _{m=2}^{n-2}{c}_m{c}_{n-m},$$ $n\ge 4$.

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Symbols:

$n$: integer, $m$: integer and $c_n$

A&S Ref:

18.5.3

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E5

Encodings:

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See also:

Annotations for §23.9 and Ch.23

►Explicit coefficients $c_n$ in terms of $c_2$ and $c_3$ are given up to $c_{19}$ in Abramowitz and Stegun (1964, p. 636). ►For $j=1,2,3$, and with $e_j$ as in §23.3(i), … ►

23.9.8 $$a_{m,n}=3(m+1)a_{m+1,n-1}+\frac{16}{3}(n+1)a_{m-2,n+1}-\frac{1}{3}(2m+3n-1)(4m+6n-1)a_{m-1,n}.$$

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Symbols:

$n$: integer, $m$: integer and $a_{m,n}$: coefficients

A&S Ref:

18.5.8

Permalink:

http://dlmf.nist.gov/23.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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19: Bibliography B

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Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 5, 109–117

External Links:

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

Referenced by:

§10.77(iv).

Encodings:

BibTeX

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Algorithms for evaluating spherical Bessel functions in the complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 28 (12), pp. 1779–1788, 1918.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 6, 122–128

External Links:

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

Referenced by:

§10.77(iv).

Encodings:

BibTeX

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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Some Geometric Aspects of Wave Motion: Wavefront Dislocations, Diffraction Catastrophes, Diffractals. In Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Vol. 36, pp. 13–28.

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External Links:

MathReview (I. Stewart), ZentralBlatt

Referenced by:

§36.14(i).

Encodings:

BibTeX

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Two cold atoms in a harmonic trap. Found. Phys. 28 (4), pp. 549–559.

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External Links:

Document

Referenced by:

§12.17.

Encodings:

BibTeX

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20: 24.2 Definitions and Generating Functions

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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$28$	$-\mathrm{2\hspace{0.33em}37494\hspace{0.33em}61029}$	$870$	$-\mathrm{2.72982\hspace{0.33em}3107}$	$\mathrm{\times 10⁷}$
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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$28$	$\mathrm{12522\hspace{0.33em}59641\hspace{0.33em}40362\hspace{0.33em}98654\hspace{0.33em}68285}$
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Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

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	$k$
…
$3$	$0$	$\frac{1}{2}$	$-\frac{3}{2}$	$1$
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Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

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	$k$
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$6$	$0$	$-3$	$0$	$5$	$0$	$-3$	$1$
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$8$	$0$	$17$	$0$	$-28$	$0$	$14$	$0$	$-4$	$1$
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