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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: 19.21 Connection Formulas

… ►The complete case of $R_J$ can be expressed in terms of $R_F$ and $R_D$: … ► $R_D(x,y,z)$ is symmetric only in $x$ and $y$, but either (nonzero) $x$ or (nonzero) $y$ can be moved to the third position by using … ►

19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

►

19.21.9 $$x{R}_D(y,z,x)+y{R}_D(z,x,y)+z{R}_D(x,y,z)=3R_F(x,y,z).$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.21(i), §19.21(ii)

Permalink:

http://dlmf.nist.gov/19.21.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►Because $R_G$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\le 0$ if the variables are real, thereby avoiding cancellations when $R_G$ is calculated from $R_F$ and $R_D$ (see §19.36(i)). …

3: 19.27 Asymptotic Approximations and Expansions

… ►

§19.27(iv) $R_D(x,y,z)$

… ►

19.27.7 $$R_D(x,y,z)=\frac{3}{2z^{3/2}}\left(\ln \left(\frac{8z}{a+g}\right)-2\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.8 $$R_D(x,y,z)=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}{R}_G(x,y,0)\left(1+O\left(\frac{z}{g}\right)\right),$$ $z/g\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $g$

Referenced by:

§19.20(iv), §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.27.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.9 $$R_D(x,y,z)=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+O\left(\frac{b}{x}\ln \frac{x}{b}\right)\right),$$ $b/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function and $b$

Permalink:

http://dlmf.nist.gov/19.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.10 $$R_D(x,y,z)=R_D(0,y,z)-\frac{3\sqrt{x}}{hz}\left(1+O\left(\sqrt{\frac{x}{h}}\right)\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $h$

Permalink:

http://dlmf.nist.gov/19.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

…

4: 19.20 Special Cases

… ►

§19.20(iv) $R_D(x,y,z)$

►

$$R_D(x,x,x)=x^{-3/2},$$

►

$$R_D(\lambda x,\lambda y,\lambda z)={\lambda }^{-3/2}{R}_D(x,y,z),$$

►

$$R_D(0,y,y)=\frac{3}{4}\pi y^{-3/2},$$

… ►

19.20.22 $${\int }_0^1\frac{t^{2}dt}{\sqrt{1-t^4}}=\frac{1}{3}{R}_D(0,2,1)=\frac{{\left(\mathrm{\Gamma }\left(\frac{3}{4}\right)\right)}^2}{{(2\pi )}^{1/2}}=\mathrm{0.59907\hspace{0.33em}01173\hspace{0.33em}67796\hspace{0.33em}10371}\mathrm{\dots }.$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Notes:

For more digits see OEIS Sequence A085566; see also Sloane (2003).

Referenced by:

§19.20(iv), §19.21(i), Figure 19.3.6, Figure 19.3.6, Figure 19.3.6

Permalink:

http://dlmf.nist.gov/19.20.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

…

5: 19.34 Mutual Inductance of Coaxial Circles

… ►

$a_3=h^2+a^2+b^2,$

… ►

19.34.3 $$2abI({𝐞}_5)=a_{3}I(\mathrm{𝟎})-I({𝐞}_3)=a_{3}I(\mathrm{𝟎})-r_{+}^2{r}_{-}^{2}I(-{𝐞}_3)=2ab(I(\mathrm{𝟎})-r_{-}^{2}I({𝐞}_1-{𝐞}_3)),$$

ⓘ

Symbols:

$I(𝐦)$: integral and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

… ►

19.34.4 $$r_{\pm }^2=a_3\pm 2ab=h^2+{(a\pm b)}^2$$

ⓘ

Symbols:

$h$: distance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

… ►Application of (19.29.4) and (19.29.7) with $\alpha =1$, $a_{\beta }+b_{\beta }t=1-t$, $\delta =3$, and $a_{\gamma }+b_{\gamma }t=1$ yields ►

19.34.5 $$\frac{3c^2}{8\pi ab}M=3R_F(0,r_{+}^2,r_{-}^2)-2r_{-}^2{R}_D(0,r_{+}^2,r_{-}^2),$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

…

6: 19.25 Relations to Other Functions

… ►Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of $R_D$; see (19.21.7). … ►In (19.25.38) and (19.25.39) $j$, $k$, $\mathrm{\ell }$ is any permutation of the numbers $1,2,3$. … ►in which $2{\omega }_1$ and $2{\omega }_3$ are generators for the lattice $𝕃$, ${\omega }_2=-{\omega }_1-{\omega }_3$, and ${\eta }_j=\zeta \left({\omega }_j\right)$ (see (23.2.12)). … ►($F_1$ and $F_D$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.) …

7: 19.28 Integrals of Elliptic Integrals

… ►

19.28.3 $${\int }_0^1{t}^{\sigma -1}(1-t)R_D(0,t,1)dt=\frac{3}{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2.$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

… ►

19.28.5 $${\int }_z^{\mathrm{\infty }}{R}_D(x,y,t)dt=6R_F(x,y,z),$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.6 $${\int }_0^1{R}_D(x,y,v^{2}z+(1-v^2)p)dv=R_J(x,y,z,p).$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.7 $${\int }_0^{\mathrm{\infty }}{R}_J(x,y,z,r^2)dr=\frac{3}{2}\pi R_F(xy,xz,yz),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

…

8: 19.17 Graphics

… ►

►►

Figure 19.17.3: $R_D(x,y,1)$ for $0\le x\le 2$, $y=0,\mathrm{\hspace{0.17em}0.1},\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}5},\mathrm{\hspace{0.17em}25}$. $y=1$ corresponds to $\frac{3}{2}(R_C(x,1)-\sqrt{x})/(1-x)$, $x\ne 1$. Magnify

… ►

►

►

Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. …The function is asymptotic to $\frac{3}{2}\pi /\sqrt{yp}$ as $p\to 0+$, and to $(\frac{3}{2}/p)\ln \left(16/y\right)$ as $y\to 0+$. …When $p=1$, it reduces to $R_D(0,y,1)$. If $y=1$, then it has the value $\frac{3}{2}\pi /(p+\sqrt{p})$ when $p>0$, and $\frac{3}{2}\pi /(p-1)$ when . … Magnify 3D Help

9: 19.16 Definitions

… ►

19.16.5 $$R_D(x,y,z)=R_J(x,y,z,z)=\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)(t+z)},$$

ⓘ

Defines:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Symbols:

$R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ , $\int$: integral and $s(t)$: scaling factor

Referenced by:

§19.18(i), §19.19, §19.20(iii), §19.20(iv), §19.24(ii), §19.6(iv)

Permalink:

http://dlmf.nist.gov/19.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

… ►and $R_D$ is a degenerate case of $R_J$, so is $R_J$ a degenerate case of the hyperelliptic integral, … ►

$b_3=a+a^{\prime }-b_1-b_2-b_4.$

… ►

19.16.15 $$R_D(x,y,z)=R_{-\frac{3}{2}}(\frac{1}{2},\frac{1}{2},\frac{3}{2};x,y,z),$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.16.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

… ►

19.16.21 $$R_D(0,y,z)=\frac{3}{4}\pi R_{-\frac{3}{2}}(\frac{1}{2},\frac{3}{2};y,z),$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\pi$: the ratio of the circumference of a circle to its diameter

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.16.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

…

10: 19.22 Quadratic Transformations

… ►

19.22.3 $$2y^2{R}_D(0,x^2,y^2)=\frac{1}{4}(y^2-x^2)R_D(0,xy,a^2)+3R_F(0,xy,a^2).$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.22(i), §19.22(i), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(i), §19.22 and Ch.19

►

19.22.4 $$(p_{\pm }^2-p_{\mp }^2)R_J(0,x^2,y^2,p^2)=2(p_{\pm }^2-a^2)R_J(0,xy,a^2,p_{\pm }^2)-3R_F(0,xy,a^2)+3\pi /(2p),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\pi$: the ratio of the circumference of a circle to its diameter and $p_{\pm }$: modified parameter

Referenced by:

§19.22(i), §19.22(i)

Permalink:

http://dlmf.nist.gov/19.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(i), §19.22 and Ch.19

… ►

19.22.10 $$R_D(0,g_0^2,a_0^2)=\frac{3\pi }{4M(a_0,g_0)a_0^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n,$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $Q_n$ , $a_n$: iterate and $g_n$: iterate

Referenced by:

§19.22(ii)

Permalink:

http://dlmf.nist.gov/19.22.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(ii), §19.22 and Ch.19

… ►

19.22.19 $$(z_{\pm }^2-z_{\mp }^2)R_D(x^2,y^2,z^2)=2(z_{\pm }^2-a^2)R_D(a^2,z_{\mp }^2,z_{\pm }^2)-3R_F(x^2,y^2,z^2)+(3/z),$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.22(iii), §19.22(iii), §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.22.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(iii), §19.22 and Ch.19

►

19.22.20 $$(p_{\pm }^2-p_{\mp }^2)R_J(x^2,y^2,z^2,p^2)=2(p_{\pm }^2-a^2)R_J(a^2,z_{+}^2,z_{-}^2,p_{\pm }^2)-3R_F(x^2,y^2,z^2)+3R_C(z^2,p^2),$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind and $p_{\pm }$: modified parameter

Referenced by:

§19.22(i), §19.22(iii), §19.22(iii), §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.22.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(iii), §19.22 and Ch.19

…