symmetric elliptic integrals

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1: 19.16 Definitions

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§19.16(i) Symmetric Integrals

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19.16.5

R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\,\mathrm{d}t}{s(t)(t+z)},

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Defines:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Symbols:: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\,\mathrm{d}\NVar{x}$ : 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

… ►which is homogeneous and of degree

-a

in the

z

’s, and unchanged when the same permutation is applied to both sets of subscripts

1,\dots,n

. … … ►

§19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

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2: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ► … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

3: 19.35 Other Applications

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4: 19.31 Probability Distributions

§19.31 Probability Distributions

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5: 19.21 Connection Formulas

§19.21 Connection Formulas

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19.21.3

6R_{G}\left(0,y,z\right)=yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))=3% zR_{F}\left(0,y,z\right)+z(y-z)R_{D}\left(0,y,z\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.21(i), §19.22(i), §19.34
Permalink:: http://dlmf.nist.gov/19.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

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19.21.11

6R_{G}\left(x,y,z\right)=3(x+y+z)R_{F}\left(x,y,z\right)-\sum x^{2}R_{D}\left(% y,z,x\right)=\sum x(y+z)R_{D}\left(y,z,x\right),

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.21(i), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.21.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

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§19.21(iii) Change of Parameter of $R_{J}$

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19.21.15

pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),

pq=yz

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.21.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

6: 19.22 Quadratic Transformations

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19.22.2

2R_{G}\left(0,x^{2},y^{2}\right)=4R_{G}\left(0,xy,a^{2}\right)-xyR_{F}\left(0,% xy,a^{2}\right),

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.22(i)
Permalink:: http://dlmf.nist.gov/19.22.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(i), §19.22 and Ch.19

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Bartky’s Transformation

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§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

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7: 19.18 Derivatives and Differential Equations

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§19.18(i) Derivatives

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19.18.2

\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x+c\right)=\tfrac{1}{2}R_{F}% \left(x+a,x+b,x+c\right).

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$
Referenced by:: §19.18(i)
Permalink:: http://dlmf.nist.gov/19.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.18(i), §19.18 and Ch.19

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§19.18(ii) Differential Equations

… ►and two similar equations obtained by permuting

x, y, z

in (19.18.10). … ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). …

8: 19.38 Approximations

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9: 19.25 Relations to Other Functions

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§19.25(i) Legendre’s Integrals as Symmetric Integrals

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§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

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§19.25(v) Jacobian Elliptic Functions

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§19.25(vii) Hypergeometric Function

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10: 19.24 Inequalities

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§19.24(i) Complete Integrals

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19.24.7

L(a,b)=8R_{G}\left(0,a^{2},b^{2}\right).

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Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $L(a,b)$ : length
Permalink:: http://dlmf.nist.gov/19.24.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.24(i), §19.24 and Ch.19

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19.24.9

\frac{1}{2}\,g_{1}^{2}\leq\frac{R_{G}\left(a_{0}^{2},g_{0}^{2},0\right)}{R_{F}% \left(a_{0}^{2},g_{0}^{2},0\right)}\leq\frac{1}{2}\,a_{1}^{2},

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.24(i), §19.9(i)
Permalink:: http://dlmf.nist.gov/19.24.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.24(i), §19.24 and Ch.19

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§19.24(ii) Incomplete Integrals

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