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

…

… ►The coefficients satisfy ►

28.14.4

{qc_{2m+2}-\left(a-(\nu+2m)^{2}\right)c_{2m}+qc_{2m-2}=0},

a=\lambda_{\nu}\left(q\right),c_{2m}=c_{2m}^{\nu}(q)

,

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter, $a$ : parameter and $c_{2m}(q)$ : coefficients
Referenced by:: item (d), item (d)
Permalink:: http://dlmf.nist.gov/28.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

… ►

28.14.5

\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Referenced by:: item (d)
Permalink:: http://dlmf.nist.gov/28.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

… ►

28.14.7

c_{-2m}^{-\nu}(q)=c_{2m}^{\nu}(q),

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

►

28.14.8

c_{2m}^{\nu}(-q)=(-1)^{m}c_{2m}^{\nu}(q).

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

…

… ►and real eigenvalues

\alpha_{1,d}

,

\alpha_{2,d}

,

\dots

,

\alpha_{d,d}

, arranged in ascending order of magnitude. … ►For

m=2

,

n=4

,

\gamma^{2}=10

, …which yields

\lambda^{2}_{4}\left(10\right)=13.97907\;345

. … ►If

\lambda^{m}_{n}\left(\gamma^{2}\right)

is known, then

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

can be found by summing (30.8.1). The coefficients

a^{m}_{n,r}(\gamma^{2})

are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). …

… ►For

n=0,1,2,3,\dots

, … ►

(a-4m^{2})B_{2m}-q(B_{2m-2}+B_{2m+2})=0

,

m=2,3,4,\dots

,

a=b_{2n+2}\left(q\right)

,

B_{2m+2}=B_{2m+2}^{2n+2}(q).

… ►

B^{2n+2}_{2n+2}(0)=1,

►

B^{2n+2}_{2m+2}(0)=0,

n\neq m

.

… ►For fixed

s=1,2,3,\dots

and fixed

m=1,2,3,\dots

, …

… ►

See accompanying text — Figure 19.3.5: $\Pi\left(\alpha^{2},k\right)$ as a function of $k^{2}$ and $\alpha^{2}$ for $-2\leq k^{2}<1$ , $-2\leq\alpha^{2}\leq 2$ . … Magnify 3D Help

►

… ►

►

…

… ►

10.53.1

\mathsf{j}_{n}\left(z\right)=z^{n}\sum_{k=0}^{\infty}\frac{(-\frac{1}{2}z^{2})% ^{k}}{k!(2n+2k+1)!!},

ⓘ

Symbols:: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.1.2
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.53 and Ch.10

►

10.53.2

\mathsf{y}_{n}\left(z\right)=-\frac{1}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!% (\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}\sum_{k=n+1}^{\infty}% \frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.

ⓘ

Symbols:: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.1.3
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.53 and Ch.10

►

10.53.3

{\mathsf{i}^{(1)}_{n}}\left(z\right)=z^{n}\sum_{k=0}^{\infty}\frac{(\frac{1}{2% }z^{2})^{k}}{k!(2n+2k+1)!!},

ⓘ

Symbols:: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.2.5
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.53 and Ch.10

►

10.53.4

{\mathsf{i}^{(2)}_{n}}\left(z\right)=\frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}% \frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{% \infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.

ⓘ

Symbols:: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.2.6
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.53 and Ch.10

►For

{\mathsf{h}^{(1)}_{n}}\left(z\right)

and

{\mathsf{h}^{(2)}_{n}}\left(z\right)

combine (10.47.10), (10.53.1), and (10.53.2). …

… ►Let

s=2m+1

,

m=0,1,2,\dots

, and

\nu

be fixed with

m<\nu<m+1

. … ►

28.16.1

\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $h$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.16 and Ch.28

…

… ►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2). ►The coefficients of the power series of

a_{2n}\left(q\right)

,

b_{2n}\left(q\right)

and also

a_{2n+1}\left(q\right)

,

b_{2n+1}\left(q\right)

are the same until the terms in

q^{2n-2}

and

q^{2n}

, respectively. … ►Here

j=1

for

a_{2n}\left(q\right)

,

j=2

for

b_{2n+2}\left(q\right)

, and

j=3

for

a_{2n+1}\left(q\right)

and

b_{2n+1}\left(q\right)

. … ►where

k

is the unique root of the equation

2E\left(k\right)=K\left(k\right)

in the interval

(0,1)

, and

k^{\prime}=\sqrt{1-k^{2}}

. … ►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2). …

… ►

28.15.1

\lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and $\nu$ : complex parameter
A&S Ref:: 20.3.15 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.15(i), §28.15 and Ch.28

… ►

28.15.2

a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.

ⓘ

Symbols:: $q=h^{2}$ : parameter, $\nu$ : complex parameter and $a$ : parameter
Referenced by:: item (e)
Permalink:: http://dlmf.nist.gov/28.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.15(i), §28.15 and Ch.28

… ►

28.15.3

\operatorname{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(% \frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}% \right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+% \frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)% ^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;

ⓘ

Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.3.17 (only two terms and without normalization)
Permalink:: http://dlmf.nist.gov/28.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.15(ii), §28.15 and Ch.28

…

… ►

p=2m+1,

… ►

29.7.4

\tau_{1}=\frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5)).

ⓘ

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §29.7(i), §29.7 and Ch.29

►The same Poincaré expansion holds for

b^{m+1}_{\nu}\left(k^{2}\right)

, since … ►

29.7.6

\tau_{2}=\frac{1}{2^{10}}(1+k^{2})(1-k^{2})^{2}(5p^{4}+34p^{2}+9),

ⓘ

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §29.7(i), §29.7 and Ch.29

… ►In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions

\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)

and

\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)

. …

布莱辛-里曼护理学院文凭办理【somewhat微aptao168】2MHDhB

11—20 of 802 matching pages

11: 34.3 Basic Properties: $\mathit{3j}$ Symbol

12: 28.14 Fourier Series

13: 30.16 Methods of Computation

14: 28.4 Fourier Series

15: 19.3 Graphics

16: 10.53 Power Series

17: 28.16 Asymptotic Expansions for Large $q$

18: 28.6 Expansions for Small $q$

19: 28.15 Expansions for Small $q$

20: 29.7 Asymptotic Expansions

布莱辛-里曼护理学院文凭办理【somewhat微aptao168】2MHDhB

11—20 of 802 matching pages

11: 34.3 Basic Properties: 3 ⁢ j Symbol

12: 28.14 Fourier Series

13: 30.16 Methods of Computation

14: 28.4 Fourier Series

15: 19.3 Graphics

16: 10.53 Power Series

17: 28.16 Asymptotic Expansions for Large q

18: 28.6 Expansions for Small q

19: 28.15 Expansions for Small q

20: 29.7 Asymptotic Expansions

11: 34.3 Basic Properties: $\mathit{3j}$ Symbol

17: 28.16 Asymptotic Expansions for Large $q$

18: 28.6 Expansions for Small $q$

19: 28.15 Expansions for Small $q$