DLMF: Untitled Document

… ►When any one of

j_{1},j_{2},j_{3}

is equal to

0,\tfrac{1}{2}

, or

1

, the

\mathit{3j}

symbol has a simple algebraic form. …For these and other results, and also cases in which any one of

j_{1},j_{2},j_{3}

is

\frac{3}{2}

or

2

, see Edmonds (1974, pp. 125–127). … ►Even permutations of columns of a

\mathit{3j}

symbol leave it unchanged; odd permutations of columns produce a phase factor

(-1)^{j_{1}+j_{2}+j_{3}}

, for example, … ►

34.3.13

\left((j_{1}+j_{2}+j_{3}+1)(-j_{1}+j_{2}+j_{3})\right)^{\frac{1}{2}}\begin{% pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{2}+m_{2})(j_{3}-m_{3})\right)^{\frac{% 1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-\frac{1}{2}\\ m_{1}&m_{2}-\frac{1}{2}&m_{3}+\frac{1}{2}\end{pmatrix}-\left((j_{2}-m_{2})(j_{% 3}+m_{3})\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-% \frac{1}{2}\\ m_{1}&m_{2}+\frac{1}{2}&m_{3}-\frac{1}{2}\end{pmatrix},

ⓘ

Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(iii), §34.3 and Ch.34

… ►

34.3.15

(2j_{1}+1)\left((j_{2}(j_{2}+1)-j_{3}(j_{3}+1))m_{1}-j_{1}(j_{1}+1)(m_{3}-m_{2% })\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(j_{1}+1)\left(j_{1}^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-j_{1}^{2}\right)^{\frac{1}{2}}% \left(j_{1}^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}-1&j_{2}&j_{% 3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}+j_{1}\left((j_{1}+1)^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-(j_{1}+1)^{2}\right)^{\frac{1}{% 2}}\left((j_{1}+1)^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}+1&j_% {2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.

ⓘ

Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(iii), §34.3 and Ch.34

…

… ►

20.8.1

\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : integer, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.8 and Ch.20

… ►This reference and Bellman (1961, pp. 46–47) include other expansions of this type.

… ► ►►

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

… ►

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},

… ►An often used alternative to the

\mathit{3j}

symbol is the Clebsch–Gordan coefficient ►

34.1.1

\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};

ⓘ

Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Sources:: Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))
Permalink:: http://dlmf.nist.gov/34.1.E1
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.15):: A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3j}$ . The sign of $m_{3}$ was changed for clarity.
See also:: Annotations for §34.1 and Ch.34

►see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)). …

… ►The identities in this section hold for

n=1,2,\dotsc

. … ►

24.6.1

B_{2n}=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{% 2n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.4

E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{% 2n},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

►

24.6.5

E_{2n}=\frac{1}{2^{n-1}}\sum_{k=0}^{n-1}(-1)^{n-k}(n-k)^{2n}\*\sum_{j=0}^{k}{2% n-2j\choose k-j}2^{j},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

►

24.6.6

E_{2n}=\sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k+1}\*\sum_{j=0}^{% \left\lfloor\tfrac{1}{2}k-\tfrac{1}{2}\right\rfloor}{k\choose j}(k-2j)^{2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

…

… ►Thus

231

is the permutation

\sigma(1)=2

,

\sigma(2)=3

,

\sigma(3)=1

. … ►Here

\sigma(1)=2,\sigma(2)=5

, and

\sigma(5)=1

. … ►A lattice path is a directed path on the plane integer lattice

\{0,1,2,\ldots\}\times\{0,1,2,\ldots\}

. … ►As an example,

\{1,3,4\}

,

\{2,6\}

,

\{5\}

is a partition of

\{1,2,3,4,5,6\}

. … ►As an example,

\{1,1,1,2,4,4\}

is a partition of 13. …

… ►

B_{2n+1}=0

,

►

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

,

n=1,2,\dots

.

… ►

E_{2n+1}=0

,

►

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

.

… ►

24.2.9

E_{n}=2^{n}E_{n}\left(\tfrac{1}{2}\right)=\text{integer},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.2
Permalink:: http://dlmf.nist.gov/24.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(ii), §24.2 and Ch.24

…

… ►

13.23.1

\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left(z+\frac{1}{2}\right)^{\mu% +\nu+\frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+% \mu+\nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right),

\Re\mu+\nu+\tfrac{1}{2}>0

,

\Re z>\tfrac{1}{2}

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $z$ : complex variable
Keywords:: Laplace transform, Mellin transform
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.23(i), §13.23 and Ch.13

… ►

13.23.8

\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}\cos\left(2xt\right)e^{-% \frac{1}{2}t^{2}}t^{-2\mu-1}M_{\kappa,\mu}\left(t^{2}\right)\,\mathrm{d}t=% \frac{\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\Gamma\left(\frac{1}{2% }+\mu+\kappa\right)}W_{\frac{1}{2}\kappa-\frac{3}{2}\mu,\frac{1}{2}\kappa+% \frac{1}{2}\mu}\left(x^{2}\right),

\Re\left(\kappa+\mu\right)>-\tfrac{1}{2}

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $x$ : real variable
Permalink:: http://dlmf.nist.gov/13.23.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §13.23(ii), §13.23 and Ch.13

… ►For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §1.16 and 3.4.42–46, 4.4.45–47, 5.94–97). … ►Let

f(x)

be absolutely integrable on the interval

[r,R]

for all positive

r<R

,

f(x)=O\left(x^{\rho_{0}}\right)

as

x\to 0+

, and

f(x)=O\left(e^{-\rho_{1}x}\right)

as

x\to+\infty

, where

\rho_{1}>\frac{1}{2}

. Then for

\mu

in the half-plane

\Re\mu\geq\mu_{1}>\max\left(-\rho_{0},\Re\kappa-\frac{1}{2}\right)

…

… ►

J. Urbanowicz (1988) On the equation

f(1)1^{k}+f(2)2^{k}+\cdots+f(x)x^{k}+{R}(x)={B}y^{2}

. Acta Arith. 51 (4), pp. 349–368.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (István Gaál), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.

ⓘ

External Links:: ISSN 0305-0041, Document, MathReview (F. I. Ibragimov), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

►

F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.

ⓘ

External Links:: ISSN 0305-0041, Document, MathReview (Roderick Wong), ZentralBlatt
Referenced by:: §14.26.
Encodings:: BibTeX

►

F. Ursell (1994) Ship Hydrodynamics, Water Waves and Asymptotics. Collected works of F. Ursell, 1946-1992, Vol. 2, World Scientific, Singapore.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §36.13.
Encodings:: BibTeX

… ►

K. M. Urwin (1964) Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.

ⓘ

External Links:: Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.31(iii), §28.31(iii).
Encodings:: BibTeX

…

… ►The set

\{2,3,4,\ldots\}

is denoted by

T

. If more than one restriction applies, then the restrictions are separated by commas, for example,

p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)

. … ►where the sum is over nonnegative integer values of

k

for which

n-\frac{1}{2}(3k^{2}\pm k)\geq 0

. … ►where the sum is over nonnegative integer values of

k

for which

n-(3k^{2}\pm k)\geq 0

. … ►where the sum is over nonnegative integer values of

m

for which

n-\tfrac{1}{2}km^{2}-m+\tfrac{1}{2}km\geq 0

. …

… ►The number of self-complementary plane partitions in

B(2r,2s,2t)

is …in

B(2r+1,2s,2t)

it is …in

B(2r+1,2s+1,2t)

it is … ►The number of symmetric self-complementary plane partitions in

B(2r,2r,2t)

is …in

B(2r+1,2r+1,2t)

it is …

重庆时时彩后2【信誉下注平台qee9.com】46E3

1—10 of 802 matching pages

1: 34.3 Basic Properties: $\mathit{3j}$ Symbol

2: 20.8 Watson’s Expansions

3: 34.1 Special Notation

4: 24.6 Explicit Formulas

5: 26.2 Basic Definitions

6: 24.2 Definitions and Generating Functions

7: 13.23 Integrals

8: Bibliography U

9: 26.10 Integer Partitions: Other Restrictions

10: 26.12 Plane Partitions

重庆时时彩后2【信誉下注平台qee9.com】46E3

1—10 of 802 matching pages

1: 34.3 Basic Properties: 3 ⁢ j Symbol

2: 20.8 Watson’s Expansions

3: 34.1 Special Notation

4: 24.6 Explicit Formulas

5: 26.2 Basic Definitions

6: 24.2 Definitions and Generating Functions

7: 13.23 Integrals

8: Bibliography U

9: 26.10 Integer Partitions: Other Restrictions

10: 26.12 Plane Partitions

1: 34.3 Basic Properties: $\mathit{3j}$ Symbol