DLMF: Untitled Document

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

… ►

R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

ⓘ

External Links:: ISSN 0893-9659, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

P. Bleher and A. Its (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

ⓘ

External Links:: ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

J. P. Buhler, R. E. Crandall, and R. W. Sompolski (1992) Irregular primes to one million. Math. Comp. 59 (200), pp. 717–722.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (Duncan A. Buell), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

H. M. Bui, B. Conrey, and M. P. Young (2011) More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

ⓘ

External Links:: ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt
Referenced by:: §25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).
Encodings:: BibTeX

… ►

P. L. Butzer, S. Flocke, and M. Hauss (1994) Euler functions

{E}_{\alpha}(z)

with complex

\alpha

and applications. In Approximation, probability, and related fields (Santa Barbara, CA, 1993), G. Anastassiou and S. T. Rachev (Eds.), pp. 127–150.

ⓘ

External Links:: MathReview (R. N. Kalia), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

…

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

for

m

up to 50 and

n

up to 25; extends Table 26.4.1 to

n=10

; tabulates Stirling numbers of the first and second kinds,

s\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts

\not\equiv\pm 2\pmod{5}

, partitions into parts

\not\equiv\pm 1\pmod{5}

, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to

\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}

. ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

.

§24.20 Tables

►Abramowitz and Stegun (1964, Chapter 23) includes exact values of

\sum_{k=1}^{m}k^{n}

,

m=1(1)100

,

n=1(1)10

;

\sum_{k=1}^{\infty}k^{-n}

,

\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}

,

\sum_{k=0}^{\infty}(2k+1)^{-n}

,

n=1,2,\dotsc

, 20D;

\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}

,

n=1,2,\dotsc

, 18D. … ►In Wagstaff (2002) these results are extended to

n=60(2)152

and

n=40(2)88

, respectively, with further complete and partial factorizations listed up to

n=300

and

n=200

, respectively. ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

… ►We use color to augment these vizualizations, either to reinforce the recognition of the height, or to convey an extra dimension to represent the phase of complex valued functions. …

§8.26 Tables

… ►

§8.26(ii) Incomplete Gamma Functions

… ►

•

Zhang and Jin (1996, Table 3.8) tabulates $\gamma\left(a,x\right)$ for $a=0.5,1,3,5,10,25,50,100$ , $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

… ►

•

Chiccoli et al. (1988) presents a short table of $E_{p}\left(x\right)$ for $p=-\tfrac{9}{2}(1)-\tfrac{1}{2}$ , $0\leq x\leq 200$ to 14S.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

§5.22 Tables

… ►

§5.22(ii) Real Variables

►Abramowitz and Stegun (1964, Chapter 6) tabulates

\Gamma\left(x\right)

,

\ln\Gamma\left(x\right)

,

\psi\left(x\right)

, and

\psi'\left(x\right)

for

x=1(.005)2

to 10D;

\psi''\left(x\right)

and

\psi^{(3)}\left(x\right)

for

x=1(.01)2

to 10D;

\Gamma\left(n\right)

,

\ifrac{1}{\Gamma\left(n\right)}

,

\Gamma\left(n+\tfrac{1}{2}\right)

,

\psi\left(n\right)

,

\operatorname{log}_{10}\Gamma\left(n\right)

,

\operatorname{log}_{10}\Gamma\left(n+\tfrac{1}{3}\right)

,

\operatorname{log}_{10}\Gamma\left(n+\tfrac{1}{2}\right)

, and

\operatorname{log}_{10}\Gamma\left(n+\tfrac{2}{3}\right)

for

n=1(1)101

to 8–11S;

\Gamma\left(n+1\right)

for

n=100(100)1000

to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates

\Gamma\left(x\right)

,

\ifrac{1}{\Gamma\left(x\right)}

,

\Gamma\left(-x\right)

,

\ln\Gamma\left(x\right)

,

\psi\left(x\right)

,

\psi\left(-x\right)

,

\psi'\left(x\right)

, and

\psi'\left(-x\right)

for

x=0(.1)5

to 8D or 8S;

\Gamma\left(n+1\right)

for

n=0(1)100(10)250(50)500(100)3000

to 51S. ►

§5.22(iii) Complex Variables

…

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_{0}\left(x\right)$ to 15D, $J_{1}\left(x\right)$ , $J_{2}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ to 10D, $Y_{2}\left(x\right)$ to 8D, $x=0(.1)17.5$ ; $Y_{n}\left(x\right)-(2/\pi)J_{n}\left(x\right)\ln x$ , $n=0,1$ , $x=0(.1)2$ , 8D; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)20$ , 5D or 5S; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 10S; modulus and phase functions $\sqrt{x}M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)-x$ , $n=0,1,2$ , $\ifrac{1}{x}=0(.01)0.1$ , 8D.

… ►

•

Döring (1971) tabulates the first 100 values of $\nu$ $(>1)$ for which $J_{-\nu}'\left(x\right)$ has the double zero $x=\nu$ , 10D.

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.1)10(.2)20$ , 8D–10D or 10S; $\sqrt{x}e^{-x}I_{n}\left(x\right)$ , $(\sqrt{x}/\pi)$ $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $1/x=0(.002)0.05$ ; $K_{0}\left(x\right)+I_{0}\left(x\right)\ln x$ , $x(K_{1}\left(x\right)-I_{1}\left(x\right)\ln x)$ , $x=0(.1)2$ , 8D; $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)10(.5)20$ , 5S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 9–10S.

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 10) give $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ $n=0(1)8$ , $x=0(.1)10$ , 5–8S; $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ $n=0(1)20(10)50$ , 100, $x=1,2,5,10,50,100$ , 10S; ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.1)5$ , 4–9D; ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $n=0(1)20(10)50$ , 100, $x=1,2,5,10,50,100$ , 10S. (For the notation see §10.1 and §10.47(ii).)

►

•

Zhang and Jin (1996, pp. 296–305) tabulates $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{j}_{n}'\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ , $\mathsf{y}_{n}'\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $\mathsf{k}_{n}'\left(x\right)$ , $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; $x\mathsf{j}_{n}\left(x\right)$ , $(x\mathsf{j}_{n}\left(x\right))^{\prime}$ , $x\mathsf{y}_{n}\left(x\right)$ , $(x\mathsf{y}_{n}\left(x\right))^{\prime}$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; real and imaginary parts of $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{j}_{n}'\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , $\mathsf{y}_{n}'\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(z\right)$ , $\mathsf{k}_{n}\left(z\right)$ , $\mathsf{k}_{n}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $z=4+2i$ , $20+10i$ , 8S. (For the notation replace $j, y, i, k$ by $\mathsf{j}$ , $\mathsf{y}$ , ${\mathsf{i}^{(1)}}$ , $\mathsf{k}$ , respectively.)

…

… ►

Subsection 25.10(ii)

In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

… ►

Subsection 3.5(vi)

Clarifications were made to this subsection with the addition of Equations (3.5.30_5), (3.5.33_1), (3.5.33_2), (3.5.33_3) and Table 3.5.17_5.

… ►

Table 3.5.19

The correct headings for the second and third columns of this table are $J_{0}\left(t\right)$ and $g(t)$ , respectively. Previously these columns were mislabeled as $g(t)$ and $J_{0}\left(t\right)$ .

$t$	$J_{0}(t)$	$g(t)$
0.0	1.00000 00000	1.00000 00000
0.5	0.93846 98072	0.93846 98072
1.0	0.76519 76866	0.76519 76865
2.0	0.22389 07791	0.22389 10326
5.0	$-$ 0.17759 67713	$-$ 0.17902 54097
10.0	$-$ 0.24593 57645	$-$ 0.07540 53543

Reported 2014-01-31 by Masataka Urago.

… ►

Table 26.8.1

Originally the Stirling number $s\left(10,6\right)$ was given incorrectly as 6327. The correct number is 63273.

$n$	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
10	$0$	$-3\,62880$	$10\,26576$	$-11\,72700$	$7\,23680$	$-2\,69325$	$63273$	$-9450$	$870$	$-45$	$1$

Reported 2013-11-25 by Svante Janson.

… ►

Table 18.3.1

Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

…

§6.19 Tables

… ►Lebedev and Fedorova (1960) and Fletcher et al. (1962) give comprehensive indexes of mathematical tables. This section lists relevant tables that appeared later. … ►

•

Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

… ►

•

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

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1—10 of 202 matching pages

1: 25.10 Zeros

2: Bibliography B

3: 26.21 Tables

§26.21 Tables

4: 24.20 Tables

§24.20 Tables

5: About Color Map

6: 8.26 Tables

§8.26 Tables

§8.26(ii) Incomplete Gamma Functions

7: 5.22 Tables

§5.22 Tables

§5.22(ii) Real Variables

§5.22(iii) Complex Variables

8: 10.75 Tables

9: Errata

10: 6.19 Tables

§6.19 Tables