DLMF: Untitled Document

… ►

B_{2n+1}=0

,

►

(-1)^{n+1}B_{2n}>0

,

n=1,2,\dots

.

… ►

E_{2n+1}=0

,

►

(-1)^{n}E_{2n}>0

.

… ►

\widetilde{E}_{n}\left(x\right)=E_{n}\left(x\right)

,

0\leq x<1

,

…

… ►

G\left(1\right)=1,

… ►

5.17.3

G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).

ⓘ

Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.17 and Ch.5

►

5.17.4

\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\,\mathrm{d}t.

ⓘ

Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.17 and Ch.5

… ►

5.17.5

\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.

ⓘ

Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $B_{\NVar{n}}$ : Bernoulli numbers, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $z$ : complex variable and $A$ : Glaisher’s constant
Referenced by:: §5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)
Permalink:: http://dlmf.nist.gov/5.17.E5
Encodings:: pMML, png, TeX
Errata (effective with 1.1.11):: For consistency we have replaced $\operatorname{Ln}z$ by $\ln z$ .
Errata (effective with 1.0.7):: Originally the term $z\operatorname{Ln}\Gamma\left(z+1\right)$ was incorrectly stated as $z\Gamma\left(z+1\right)$ . (This error was reported subsequently by Nick Jones on December 11, 2013.)
Reported 2013-08-01 by Gergő Nemes
See also:: Annotations for §5.17 and Ch.5

… ►

5.17.7

C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1% }{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\gamma+\ln\left% (2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'% \left(-1\right),

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer, $k$ : nonnegative integer and $C$ : log of Glaisher’s Constant
Referenced by:: §5.17
Permalink:: http://dlmf.nist.gov/5.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.17 and Ch.5

…

… ►For

k=0,1

, the multinomial coefficient is defined to be

1

. … ►

M_{1}

is the multinominal coefficient (26.4.2): …

M_{2}

is the number of permutations of

\{1,2,\ldots,n\}

with

a_{1}

cycles of length 1,

a_{2}

cycles of length 2,

\dotsc

, and

a_{n}

cycles of length

n

: …

M_{3}

is the number of set partitions of

\{1,2,\ldots,n\}

with

a_{1}

subsets of size 1,

a_{2}

subsets of size 2,

\dotsc

, and

a_{n}

subsets of size

n

: …For each

n

all possible values of

a_{1},a_{2},\ldots,a_{n}

are covered. …

… ►

D(m,n)

is the number of paths from

(0,0)

to

(m,n)

that are composed of directed line segments of the form

(1,0)

,

(0,1)

, or

(1,1)

. … ►

M(n)

is the number of lattice paths from

(0,0)

to

(n,n)

that stay on or above the line

y=x

and are composed of directed line segments of the form

(2,0)

,

(0,2)

, or

(1,1)

. … ►

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)

. … ►

r(n)

is the number of paths from

(0,0)

to

(n,n)

that stay on or above the diagonal

y=x

and are composed of directed line segments of the form

(1,0)

,

(0,1)

, or

(1,1)

. … ►

26.6.10

D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),

m,n\geq 1

,

ⓘ

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer and $D(m,n)$ : Delannoy number
Referenced by:: §26.6(iii)
Permalink:: http://dlmf.nist.gov/26.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iii), §26.6 and Ch.26

…

… ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates

T_{n}\left(x\right)

,

U_{n}\left(x\right)

,

L_{n}\left(x\right)

, and

H_{n}\left(x\right)

for

n=0(1)12

. The ranges of

x

are

0.2(.2)1

for

T_{n}\left(x\right)

and

U_{n}\left(x\right)

, and

0.5,1,3,5,10

for

L_{n}\left(x\right)

and

H_{n}\left(x\right)

. …

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

… ►

W. Groenevelt (2007) Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

ⓘ

External Links:: ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt
Referenced by:: §18.28(xi), §18.38(iii).
Encodings:: BibTeX

… ►

V. I. Gromak (1976) The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 12(4), pp. 519–521 (1977)
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(v).
Encodings:: BibTeX

►

V. I. Gromak (1978) One-parameter systems of solutions of Painlevé equations. Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 14(12), pp. 1510–1513 (1979)
External Links:: ISSN 0374-0641, MathReview (M. Tvrdý), ZentralBlatt
Referenced by:: §32.10(i), §32.10(ii), §32.10(iii), §32.7(iv).
Encodings:: BibTeX

… ►

J. H. Gunn (1967) Algorithm 300: Coulomb wave functions. Comm. ACM 10 (4), pp. 244–245.

ⓘ

Notes:: For remarks see same journal 12(1969), p. 692 and 16(1973), pp. 308-309 and for certification see same journal 12(1969), pp. 279-280.
External Links:: ISSN 0001-0782, Document
Referenced by:: §33.26(ii).
Encodings:: BibTeX

…

… ►The conjugate to the example in Figure 26.9.1 is

6+5+4+2+1+1+1

. … ►It is also equal to the number of lattice paths from

(0,0)

to

(m,k)

that have exactly

n

vertices

(h,j)

,

1\leq h\leq m

,

1\leq j\leq k

, above and to the left of the lattice path. … ►

p_{3}\left(n\right)=1+\left\lfloor\frac{n^{2}+6n}{12}\right\rfloor.

… ►It is also assumed everywhere that

|q|<1

. … ►Also, when

|xq|<1

…

… ►Results of low (

~{}2

to

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

to

20

. … ►Equation (18.40.7) provides step-histogram approximations to

\int_{a}^{x}\,\mathrm{d}\mu(x)

, as shown in Figure 18.40.1 for

N=12

and

120

, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. … ►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in

\mu(x)

, using what is referred to as the Stieltjes derivative which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). … ►where the coefficients are defined recursively via

a_{1}=\frac{x_{1,N}}{x_{2,N}}-1

, and … ►This is a challenging case as the desired

w^{\mathrm{RCP}}(x)

on

[-1,1]

has an essential singularity at

x=-1

. …

… ►Cody and Hillstrom (1967) gives minimax rational approximations for

\ln\Gamma\left(x\right)

for the ranges

0.5\leq x\leq 1.5

,

1.5\leq x\leq 4

,

4\leq x\leq 12

; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to

\Gamma\left(x\right)

and

\ln\Gamma\left(x\right)

in the intervals

0\leq x\leq 1

,

8\leq x\leq 1000

,

12\leq x\leq 1000

; precision is variable. … ►Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of

\Gamma\left(1+x\right)

,

1/\Gamma\left(1+x\right)

,

\Gamma\left(x+3\right)

,

\ln\Gamma\left(x+3\right)

,

\psi\left(x+3\right)

, and the first six derivatives of

\psi\left(x+3\right)

for

0\leq x\leq 1

. …Clenshaw (1962) also gives 20D Chebyshev-series coefficients for

\Gamma\left(1+x\right)

and its reciprocal for

0\leq x\leq 1

. …

… ►

23.14.2

\int{\wp}^{2}\left(z\right)\,\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1% }{12}g_{2}z,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.1
Permalink:: http://dlmf.nist.gov/23.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

►

23.14.3

\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.2
Permalink:: http://dlmf.nist.gov/23.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

►For further integrals see Gröbner and Hofreiter (1949, Vol. 1, pp. 161–162), Gradshteyn and Ryzhik (2000, p. 622), and Prudnikov et al. (1990, pp. 51–52).

.世界杯各队主教练年薪『网址:mxsty.cc』.世界杯季军怎么算-m6q3s2-2022年12月1日13时59分20秒

21—30 of 837 matching pages

21: 24.2 Definitions and Generating Functions

22: 5.17 Barnes’ $G$ -Function (Double Gamma Function)

23: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

24: 26.6 Other Lattice Path Numbers

25: 18.41 Tables

26: Bibliography G

27: 26.9 Integer Partitions: Restricted Number and Part Size

28: 18.40 Methods of Computation

29: 5.23 Approximations

30: 23.14 Integrals

.世界杯各队主教练年薪『网址:mxsty.cc』.世界杯季军怎么算-m6q3s2-2022年12月1日13时59分20秒

21—30 of 837 matching pages

21: 24.2 Definitions and Generating Functions

22: 5.17 Barnes’ G -Function (Double Gamma Function)

23: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

24: 26.6 Other Lattice Path Numbers

25: 18.41 Tables

26: Bibliography G

27: 26.9 Integer Partitions: Restricted Number and Part Size

28: 18.40 Methods of Computation

29: 5.23 Approximations

30: 23.14 Integrals

22: 5.17 Barnes’ $G$ -Function (Double Gamma Function)