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1: 34.2 Definition: $\mathit{3}j$ Symbol

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

►The quantities $j_1,j_2,j_3$ in the $\mathit{3}j$ symbol are called angular momenta. …They therefore satisfy the triangle conditions …where $r,s,t$ is any permutation of $1,2,3$. The corresponding projective quantum numbers $m_1,m_2,m_3$ are given by …

2: 22.9 Cyclic Identities

… ► … ►

22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2).$$

ⓘ

Symbols:

$d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

§22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

… ►

§22.9(iii) Typical Identities of Rank 3

… ►

22.9.22 $$s_{1,3}^{(2)}{c}_{1,3}^{(2)}{d}_{2,3}^{(2)}{d}_{3,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}{d}_{3,3}^{(2)}{d}_{1,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}{d}_{1,3}^{(2)}{d}_{2,3}^{(2)}=\frac{{\kappa }^2+k^2-1}{1-{\kappa }^2}\left(s_{1,3}^{(2)}{c}_{1,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}\right),$$

ⓘ

Symbols:

$k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

►

22.9.23 $$s_{1,3}^{(4)}{d}_{1,3}^{(4)}{c}_{2,3}^{(4)}{c}_{3,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}{c}_{3,3}^{(4)}{c}_{1,3}^{(4)}+s_{3,3}^{(4)}{d}_{3,3}^{(4)}{c}_{1,3}^{(4)}{c}_{2,3}^{(4)}=\frac{{\kappa }^2}{1-{\kappa }^2}\left(s_{1,3}^{(4)}{d}_{1,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}\right).$$

ⓘ

Symbols:

$s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

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

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

… ►

§34.3(ii) Symmetry

… ►

§34.3(iv) Orthogonality

… ►

§34.3(vi) Sums

… ►

4: 9.10 Integrals

… ►

${\displaystyle {\int }_0^{\mathrm{\infty }}}\mathrm{Ai}\left(t\right)dt=\frac{1}{3}$ ,

►

${\displaystyle {\int }_{-\mathrm{\infty }}^0}\mathrm{Ai}\left(t\right)dt=\frac{2}{3}$ ,

… ►

9.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{-p^3/3}\left(\frac{1}{3}-\frac{p{}_1{}^{}F_1^{}(\frac{1}{3};\frac{4}{3};\frac{1}{3}{p}^3)}{3^{4/3}\mathrm{\Gamma }\left(\frac{4}{3}\right)}+\frac{p^2{}_1{}^{}F_1^{}(\frac{2}{3};\frac{5}{3};\frac{1}{3}{p}^3)}{3^{5/3}\mathrm{\Gamma }\left(\frac{5}{3}\right)}\right),$$ $p\in ℂ$.

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

►

9.10.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(-t\right)dt=\frac{1}{3}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}+\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right),$$ $\mathrm{\Re }p>0$,

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

… ►For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).

5: 4.43 Cubic Equations

… ►

$A={\left(-\frac{4}{3}p\right)}^{1/2},$

… ►

4.43.2 $$z^3+pz+q=0$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

… ►

$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

►

$A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

►

$B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

…

6: 36.9 Integral Identities

… ►

36.9.1 $${|{\mathrm{\Psi }}_1\left(x\right)|}^2=2^{5/3}{\int }_0^{\mathrm{\infty }}{\mathrm{\Psi }}_1\left(2^{2/3}(3u^2+x)\right)du;$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

… ►

36.9.2 $${(\mathrm{Ai}\left(x\right))}^2=\frac{2^{2/3}}{\pi }{\int }_0^{\mathrm{\infty }}\mathrm{Ai}\left(2^{2/3}(u^2+x)\right)du.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $x$: real parameter

Referenced by:

§9.5(i)

Permalink:

http://dlmf.nist.gov/36.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

… ►

36.9.4 $${|{\mathrm{\Psi }}_2\left(x,y\right)|}^2={\int }_0^{\mathrm{\infty }}\left({\mathrm{\Psi }}_1\left(\frac{4u^3+2uy+x}{u^{1/3}}\right)+{\mathrm{\Psi }}_1\left(\frac{4u^3+2uy-x}{u^{1/3}}\right)\right)\frac{du}{u^{1/3}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

… ►

36.9.8 $${\left|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,z\right)\right|}^2=8{\pi }^2{\left(\frac{2}{9}\right)}^{1/3}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{Ai}\left({\left(\frac{4}{3}\right)}^{1/3}(x+zv+3u^2)\right)\mathrm{Ai}\left({\left(\frac{4}{3}\right)}^{1/3}(y+zu+3v^2)\right)dudv.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.9 $${\left|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right)\right|}^2=\frac{8{\pi }^2}{3^{2/3}}{\int }_0^{\mathrm{\infty }}{\int }_0^{2\pi }\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+\mathrm{i}y+2zu\exp \left(\mathrm{i}\theta \right)+3u^2\exp \left(-2\mathrm{i}\theta \right)\right)\right)\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-\mathrm{i}y+2zu\exp \left(-\mathrm{i}\theta \right)+3u^2\exp \left(2\mathrm{i}\theta \right)\right)\right)\right)udud\theta .$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, integral identity

Permalink:

http://dlmf.nist.gov/36.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

…

7: 9.5 Integral Representations

… ►

9.5.2 $$\mathrm{Ai}\left(-x\right)=\frac{x^{1/2}}{\pi }{\int }_{-1}^{\mathrm{\infty }}\cos \left(x^{3/2}(\frac{1}{3}{t}^3+t^2-\frac{2}{3})\right)dt,$$ $x>0$.

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Source:

Olver (1997b, p. 103)

A&S Ref:

10.4.32 (in different form)

Permalink:

http://dlmf.nist.gov/9.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(i), §9.5 and Ch.9

… ►

9.5.4 $$\mathrm{Ai}\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }{\mathrm{e}}^{-\pi \mathrm{i}/3}}^{\mathrm{\infty }{\mathrm{e}}^{\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt,$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Source:

Olver (1997b, p. 53)

Referenced by:

§36.2(ii), §9.14, §9.17(iii), (9.5.5)

Permalink:

http://dlmf.nist.gov/9.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

►

9.5.5 $$\mathrm{Bi}\left(z\right)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }{\mathrm{e}}^{\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt+\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }{\mathrm{e}}^{-\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt.$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.2.10) and (9.5.4).

Referenced by:

(9.5.3)

Permalink:

http://dlmf.nist.gov/9.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

►

9.5.6 $$\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{t^3}{3}-\frac{z^3}{3t^3}\right)dt,$$ .

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Source:

Reid (1995, (5.4), p. 170, with change of variable and using analytic continuation to complex $z$)

Referenced by:

Erratum (V1.0.15) for Equation (9.5.6)

Permalink:

http://dlmf.nist.gov/9.5.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

The validity constraint was added.

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

… ►In (9.5.7) and (9.5.8) $\zeta =\frac{2}{3}{z}^{3/2}$. …

8: 1.11 Zeros of Polynomials

… ►Set $z=w-\frac{1}{3}a$ to reduce $f(z)=z^3+a{z}^2+bz+c$ to $g(w)=w^3+pw+q$, with $p=(3b-a^2)/3$, $q=(2a^3-9ab+27c)/27$. … ►Addition of $-\frac{1}{3}a$ to each of these roots gives the roots of $f(z)=0$. … ► $f(z)=z^3-6z^2+6z-2$, $g(w)=w^3-6w-6$, $A=3\sqrt[3]{4}$, $B=3\sqrt[3]{2}$. Roots of $f(z)=0$ are $2+\sqrt[3]{4}+\sqrt[3]{2}$, $2+\sqrt[3]{4}\rho +\sqrt[3]{2}{\rho }^2$, $2+\sqrt[3]{4}{\rho }^2+\sqrt[3]{2}\rho$. … ►Set $z=w-\frac{1}{4}a$ to reduce $f(z)=z^4+a{z}^3+b{z}^2+cz+d$ to …

9: 19.32 Conformal Map onto a Rectangle

… ►with $x_1,x_2,x_3$ real constants, has differential ►

19.32.2 $$dz=-\frac{1}{2}\left(\prod _{j=1}^3{(p-x_j)}^{-1/2}\right)dp,$$ $\mathrm{\Im }p>0$; , $j=1,2,3$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase and $z(p)$: function

Referenced by:

§19.32

Permalink:

http://dlmf.nist.gov/19.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.32 and Ch.19

… ►

$$z(x_2)=z(x_1)+z(x_3),$$

►

$$z(x_3)=R_F(x_3-x_1,x_3-x_2,0)=-\mathrm{i}{R}_F(0,x_1-x_3,x_2-x_3).$$

►As $p$ proceeds along the entire real axis with the upper half-plane on the right, $z$ describes the rectangle in the clockwise direction; hence $z(x_3)$ is negative imaginary. …

10: 22.13 Derivatives and Differential Equations

… ►

22.13.13 $$\frac{d^2}{{dz}^2}\mathrm{sn}(z,k)=-(1+k^2)\mathrm{sn}(z,k)+2k^2{\mathrm{sn}}^3(z,k),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex and $k$: modulus

Referenced by:

§22.10(i), §22.10(ii), §22.13(iii)

Permalink:

http://dlmf.nist.gov/22.13.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

►

22.13.14 $$\frac{d^2}{{dz}^2}\mathrm{cn}(z,k)=-(k_{}^{\prime }{}_{}{}^2-k^2)\mathrm{cn}(z,k)-2k^2{\mathrm{cn}}^3(z,k),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Referenced by:

§22.10(i), §22.10(ii)

Permalink:

http://dlmf.nist.gov/22.13.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

►

22.13.15 $$\frac{d^2}{{dz}^2}\mathrm{dn}(z,k)=(1+k_{}^{\prime }{}_{}{}^2)\mathrm{dn}(z,k)-2{\mathrm{dn}}^3(z,k).$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Referenced by:

§22.10(i), §22.10(ii)

Permalink:

http://dlmf.nist.gov/22.13.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

►

22.13.16 $$\frac{d^2}{{dz}^2}\mathrm{cd}(z,k)=-(1+k^2)\mathrm{cd}(z,k)+2k^2{\mathrm{cd}}^3(z,k),$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

►

22.13.17 $$\frac{d^2}{{dz}^2}\mathrm{sd}(z,k)=(k^2-k_{}^{\prime }{}_{}{}^2)\mathrm{sd}(z,k)-2k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^3(z,k),$$

ⓘ

Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

…