permutations

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21: 34.2 Definition: $\mathit{3}j$ Symbol

… ►where $r,s,t$ is any permutation of $1,2,3$. …

22: 19.25 Relations to Other Functions

… ►then the five nontrivial permutations of $x,y,z$ that leave $R_F$ invariant change $k^2$ ($=(z-y)/(z-x)$) into $1/k^2$, $k_{}^{\prime }{}_{}{}^2$, $1/k_{}^{\prime }{}_{}{}^2$, $-k^2/k_{}^{\prime }{}_{}{}^2$, $-k_{}^{\prime }{}_{}{}^2/k^2$, and $\sin \varphi$ ($=\sqrt{(z-x)/z}$) into $k\sin \varphi$, $-\mathrm{i}\tan \varphi$, $-\mathrm{i}{k}^{\prime }\tan \varphi$, $(k^{\prime }\sin \varphi )/\sqrt{1-k^2{\sin }^2\varphi }$, $-\mathrm{i}k\sin \varphi /\sqrt{1-k^2{\sin }^2\varphi }$. Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … ►With $0\le k^2\le 1$ and $\mathrm{p},\mathrm{q},\mathrm{r}$ any permutation of the letters $\mathrm{c},\mathrm{d},\mathrm{n}$, define … ►In (19.25.38) and (19.25.39) $j$, $k$, $\mathrm{\ell }$ is any permutation of the numbers $1,2,3$. …

23: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►Even permutations of columns of a $\mathit{3}j$ symbol leave it unchanged; odd permutations of columns produce a phase factor ${(-1)}^{j_1+j_2+j_3}$, for example, …

24: 19.18 Derivatives and Differential Equations

… ►and two similar equations obtained by permuting $x,y,z$ in (19.18.10). …

25: 1.6 Vectors and Vector-Valued Functions

… ►

1.6.14

ⓘ

Defines:

${ϵ}_{jk\mathrm{\ell }}$: Levi-Civita symbol

Symbols:

$j$: integer and $k$: integer

Permalink:

http://dlmf.nist.gov/1.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

…

26: 19.20 Special Cases

… ►where $x,y,z$ may be permuted. … ►If $x,y,z$ are permuted so that , then the Cauchy principal value of $R_J$ is given by …

27: 19.29 Reduction of General Elliptic Integrals

… ►and $\alpha ,\beta ,\gamma ,\delta$ is any permutation of the numbers $1,2,3,4$, then … ►where $\alpha ,\beta ,\gamma$ is any permutation of the numbers $1,2,3$, and …

28: 22.6 Elementary Identities

… ►If $\{\text{p,q,r}\}$ is any permutation of $\{\text{c,d,n}\}$, then …

29: 23.10 Addition Theorems and Other Identities

… ►(23.10.8) continues to hold when $e_1$, $e_2$, $e_3$ are permuted cyclically. …

30: 31.2 Differential Equations

… ►There are $4!=24$ homographies $\tilde{z}(z)=(Az+B)/(Cz+D)$ that take $0,1,a,\mathrm{\infty }$ to some permutation of $0,1,a^{\prime },\mathrm{\infty }$, where $a^{\prime }$ may differ from $a$. …