symmetric elliptic%0Aintegrals

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21: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

►For $N=0,1,2,\mathrm{\dots }$ define the homogeneous hypergeometric polynomial … ►Define the elementary symmetric function $E_s(𝐳)$ by … ►The number of terms in $T_N$ can be greatly reduced by using variables $𝐙=\mathrm{𝟏}-(𝐳/A)$ with $A$ chosen to make $E_1(𝐙)=0$. … ►

22: 19.34 Mutual Inductance of Coaxial Circles

§19.34 Mutual Inductance of Coaxial Circles

… ►

$a_5=0,$

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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

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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

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Annotations for §19.34 and Ch.19

… ►References for other inductance problems solvable in terms of elliptic integrals are given in Grover (1946, pp. 8 and 283).

23: 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 … ►

24: 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

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Annotations for §19.32 and Ch.19

… ►

$$z(\mathrm{\infty })=0,$$

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$$z(x_1)=R_F(0,x_1-x_2,x_1-x_3)\text{(}>0\text{)},$$

… ►For further connections between elliptic integrals and conformal maps, see Bowman (1953, pp. 44–85).

25: 19.28 Integrals of Elliptic Integrals

§19.28 Integrals of Elliptic Integrals

►In (19.28.1)–(19.28.3) we assume $\mathrm{\Re }\sigma >0$. … ►

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

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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

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Annotations for §19.28 and Ch.19

… ►

26: 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

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

… ►

§19.33(ii) Potential of a Charged Conducting Ellipsoid

… ►and the electric capacity $C=Q/V(0)$ is given by … ►A conducting elliptic disk is included as the case $c=0$. … ►Ellipsoidal distributions of charge or mass are used to model certain atomic nuclei and some elliptical galaxies. …

27: 19.7 Connection Formulas

§19.7 Connection Formulas

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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Imaginary-Argument Transformation

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§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

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28: 19.17 Graphics

§19.17 Graphics

►See Figures 19.17.1–19.17.8 for symmetric elliptic integrals with real arguments. … ►For $R_F$, $R_G$, and $R_J$, which are symmetric in $x,y,z$, we may further assume that $z$ is the largest of $x,y,z$ if the variables are real, then choose $z=1$, and consider only $0\le x\le 1$ and $0\le y\le 1$. The cases $x=0$ or $y=0$ correspond to the complete integrals. … ►To view $R_F(0,y,1)$ and $2R_G(0,y,1)$ for complex $y$, put $y=1-k^2$, use (19.25.1), and see Figures 19.3.7–19.3.12. …

29: 20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

… ►In the case of the symmetric integrals, with the notation of §19.16(i) we have … ►

§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. … ►As a function of $\tau$, $k^2$ is the elliptic modular function; see Walker (1996, Chapter 7) and (23.15.2), (23.15.6). …

30: Bille C. Carlson

… ►If some of the parameters are equal, then the $R$-function is symmetric in the corresponding variables. 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. Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. … ►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. … ►

19 Elliptic Integrals

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