DLMF: Untitled Document

… ►

Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

.

► ►►►

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
0	0	1	0	$\infty$	1	$\infty$
…

► ►

4.17.1

\lim_{z\to 0}\frac{\sin z}{z}=1,

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.74
Permalink:: http://dlmf.nist.gov/4.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

►

4.17.2

\lim_{z\to 0}\frac{\tan z}{z}=1.

ⓘ

Symbols:: $\tan\NVar{z}$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.75
Permalink:: http://dlmf.nist.gov/4.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

►

4.17.3

\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

… ►where

k=1,2,\dots,n

. … ►with

\beta_{n}>0

,

p_{-1}(x)=0

, and

p_{0}(x)=1

. …with

q_{-1}(x)=0

, and

q_{0}(x)=1/\sqrt{h_{0}}

. … ►We choose

s=1

so that

f(\zeta)=O\left(1\right)

at infinity. … ►with saddle point at

t=1

, and when

c=1

the vertical path intersects the real axis at the saddle point. …

… ►In the sector

|\operatorname{ph}z|<\tfrac{1}{2}\pi

the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). On the rays

\operatorname{ph}z=\pm\tfrac{1}{2}\pi

, integration can proceed in either direction. … ►

Example 1

►We assume

2\mu\neq-1,-2,-3,\dots

. … ►We assume

a,a+1-b\neq 0,-1,-2,\dots

. …

… ►Let

\rho(x)

be a polynomial of degree

\ell

and positive when

-1\leq x\leq 1

. The Bernstein–Szegő polynomials

\{p_{n}(x)\}

,

n=0,1,\dots

, are orthogonal on

(-1,1)

with respect to three types of weight function:

(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}

,

(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}

,

(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}

. In consequence,

p_{n}(\cos\theta)

can be given explicitly in terms of

\rho(\cos\theta)

and sines and cosines, provided that

\ell<2n

in the first case,

\ell<2n+2

in the second case, and

\ell<2n+1

in the third case. …

… ►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. …

… ►If any lower argument in a

\mathit{6j}

symbol is

0

,

\tfrac{1}{2}

, or

1

, then the

\mathit{6j}

symbol has a simple algebraic form. … ►

34.5.4

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

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

… ►

34.5.11

{(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half., $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

… ►

34.5.18

\sum_{j}(-1)^{j_{1}+j_{2}+j}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{2}&j_{1}&j^{\prime}\end{Bmatrix}=\sqrt{(2j_{1}+1)(2j_{2}+1)}\,\delta_{j^{% \prime},0},

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

… ►

34.5.22

\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\frac{1}{l(l+1)}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}=\frac{1}{j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}\left(\frac{% (2j_{1}-j)!(2j_{2}+j+1)!}{(2j_{2}-j)!(2j_{1}+j+1)!}\right)^{\frac{1}{2}},

j_{2}<j_{1}

.

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

…

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

… ►Assume

J_{\nu-1}\left(z\right)\neq 0

. … ►

10.10.1

\frac{J_{\nu}\left(z\right)}{J_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}-% \cfrac{1}{2(\nu+1)z^{-1}-\cfrac{1}{2(\nu+2)z^{-1}-\cdots}}},

z\neq 0

,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.73
Referenced by:: §10.33, §10.74(v)
Permalink:: http://dlmf.nist.gov/10.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.10 and Ch.10

►

10.10.2

\frac{J_{\nu}\left(z\right)}{J_{\nu-1}\left(z\right)}=\cfrac{\tfrac{1}{2}z/\nu% }{1-\cfrac{\tfrac{1}{4}z^{2}/(\nu(\nu+1))}{1-\cfrac{\tfrac{1}{4}z^{2}/((\nu+1)% (\nu+2))}{1-\cdots}}},

\nu\neq 0,-1,-2,\dotsc

.

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.73
Referenced by:: §10.33, §10.74(v)
Permalink:: http://dlmf.nist.gov/10.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.10 and Ch.10

…

… ►Assume

I_{\nu-1}\left(z\right)\neq 0

. … ►

10.33.1

\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}+}% \cfrac{1}{2(\nu+1)z^{-1}+}\cfrac{1}{2(\nu+2)z^{-1}+}\cdots,

z\neq 0

,

ⓘ

Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.74(v)
Permalink:: http://dlmf.nist.gov/10.33.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.33 and Ch.10

►

10.33.2

\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{\frac{1}{2}z/\nu}% {1+}\cfrac{\frac{1}{4}z^{2}/(\nu(\nu+1))}{1+}\cfrac{\frac{1}{4}z^{2}/((\nu+1)(% \nu+2))}{1+}\cdots,

\nu\neq 0,-1,-2,\dotsc

.

ⓘ

Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.74(v)
Permalink:: http://dlmf.nist.gov/10.33.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.33 and Ch.10

…

… ►

34.8.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),

j_{1},j_{2},j_{3}\gg l_{3}\gg 1

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

… ►

34.8.2

\cos\theta=\frac{j_{1}(j_{1}+1)+j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}{2\sqrt{j_{1}(j_% {1}+1)j_{2}(j_{2}+1)}},

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

►and the symbol

o\left(1\right)

denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). …

👉1(716)351-6210📞Delta Flights 🪐Airlines 🛸flight cancellation

11—20 of 823 matching pages

11: 4.17 Special Values and Limits

12: 3.5 Quadrature

13: 13.29 Methods of Computation

Example 1

14: 18.31 Bernstein–Szegő Polynomials

15: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

16: 34.5 Basic Properties: $\mathit{6j}$ Symbol

17: 24.20 Tables

18: 10.10 Continued Fractions

19: 10.33 Continued Fractions

20: 34.8 Approximations for Large Parameters

👉1(716)351-6210📞Delta Flights 🪐Airlines 🛸flight cancellation

11—20 of 823 matching pages

11: 4.17 Special Values and Limits

12: 3.5 Quadrature

13: 13.29 Methods of Computation

Example 1

14: 18.31 Bernstein–Szegő Polynomials

15: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

16: 34.5 Basic Properties: 6 ⁢ j Symbol

17: 24.20 Tables

18: 10.10 Continued Fractions

19: 10.33 Continued Fractions

20: 34.8 Approximations for Large Parameters

16: 34.5 Basic Properties: $\mathit{6j}$ Symbol