symmetric elliptic integrals

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11: 19.34 Mutual Inductance of Coaxial Circles

§19.34 Mutual Inductance of Coaxial Circles

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

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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

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19.34.6 $$\frac{c^2}{2\pi }M=(r_{+}^2+r_{-}^2)R_F(0,r_{+}^2,r_{-}^2)-4R_G(0,r_{+}^2,r_{-}^2).$$

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Symbols:

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

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

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12: 19.27 Asymptotic Approximations and Expansions

§19.27 Asymptotic Approximations and Expansions

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19.27.4 $$R_G(x,y,z)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln \frac{z}{a}\right)\right),$$ $a/z\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\ln z$: principal branch of logarithm function and $a$

Permalink:

http://dlmf.nist.gov/19.27.E4

Encodings:

pMML, png, TeX

See also:

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

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

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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

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19.27.11 $$R_J(x,y,z,p)=\frac{3}{p}{R}_F(x,y,z)-\frac{3\pi }{2p^{3/2}}\left(1+O\left(\sqrt{\frac{c}{p}}\right)\right),$$ $c/p\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: 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 $c$

Referenced by:

§19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E11

Encodings:

pMML, png, TeX

See also:

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

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13: 19.1 Special Notation

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$l,m,n$	nonnegative integers.
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… ►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. … ►

$R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n).$

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14: 19.33 Triaxial Ellipsoids

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19.33.1 $$S=3V{R}_G(a^{-2},b^{-2},c^{-2}),$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $V$: volume and $S$: area

Referenced by:

§19.33(i)

Permalink:

http://dlmf.nist.gov/19.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(i), §19.33 and Ch.19

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§19.33(ii) Potential of a Charged Conducting Ellipsoid

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19.33.5 $$V(\lambda )=Q{R}_F(a^2+\lambda ,b^2+\lambda ,c^2+\lambda ),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $Q$: charge and $V(\lambda )$: potential

Permalink:

http://dlmf.nist.gov/19.33.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(ii), §19.33 and Ch.19

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19.33.6 $$1/C=R_F(a^2,b^2,c^2).$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $C$: electric capacity

Permalink:

http://dlmf.nist.gov/19.33.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(ii), §19.33 and Ch.19

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19.33.11 $$U=\frac{1}{2}{(\alpha \beta \gamma )}^2{R}_F({\alpha }^2,{\beta }^2,{\gamma }^2){\int }_0^{\mathrm{\infty }}{(g(r))}^{2}dr,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ , $\int$: integral, $\alpha$: positive constant, $\beta$: positive constant, $\gamma$: positive constant, $U$: energy and $g(r)$: function

Permalink:

http://dlmf.nist.gov/19.33.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(iv), §19.33 and Ch.19

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15: 19.28 Integrals of Elliptic Integrals

§19.28 Integrals of Elliptic Integrals

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

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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

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

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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

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

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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

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16: 19.20 Special Cases

§19.20 Special Cases

… ►The general lemniscatic case is … ►where $x,y,z$ may be permuted. … ►The general lemniscatic case is … ►

17: 19.32 Conformal Map onto a Rectangle

§19.32 Conformal Map onto a Rectangle

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19.32.1 $$z(p)=R_F(p-x_1,p-x_2,p-x_3),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $z(p)$: function

Permalink:

http://dlmf.nist.gov/19.32.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.32 and Ch.19

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18: 19.17 Graphics

§19.17 Graphics

►See Figures 19.17.1–19.17.8 for symmetric elliptic integrals with real arguments. … ►

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Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. … Magnify 3D Help

19: Bille C. Carlson

… ►This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. … ►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

20: 19.26 Addition Theorems

§19.26 Addition Theorems

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19.26.8 $$2R_G(x+\lambda ,y+\lambda ,z+\lambda )+2R_G(x+\mu ,y+\mu ,z+\mu )=2R_G(x,y,z)+\lambda R_F(x+\lambda ,y+\lambda ,z+\lambda )+\mu R_F(x+\mu ,y+\mu ,z+\mu )+\sqrt{\lambda +\mu +x+y+z}.$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\lambda$: positive, $x$: positive, $y$: positive, $z$: positive and $\mu >0$: parameter

Permalink:

http://dlmf.nist.gov/19.26.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

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§19.26(iii) Duplication Formulas

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19.26.20 $$R_D(x,y,z)=2R_D(x+\lambda ,y+\lambda ,z+\lambda )+\frac{3}{\sqrt{z}(z+\lambda )}.$$

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Symbols:

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

Referenced by:

§19.26(iii)

Permalink:

http://dlmf.nist.gov/19.26.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

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19.26.21 $$2R_G(x,y,z)=4R_G(x+\lambda ,y+\lambda ,z+\lambda )-\lambda R_F(x,y,z)-\sqrt{x}-\sqrt{y}-\sqrt{z}.$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive

Referenced by:

§19.36(i)

Permalink:

http://dlmf.nist.gov/19.26.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

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