symmetric elliptic integrals

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21: 19.23 Integral Representations

§19.23 Integral Representations

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19.23.1

R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)^% {-1/2}\,\mathrm{d}\theta,

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

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19.23.2

R_{G}\left(0,y,z\right)=\frac{1}{2}\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^% {2}\theta)^{1/2}\,\mathrm{d}\theta,

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Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Permalink:: http://dlmf.nist.gov/19.23.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

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19.23.3

R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)% ^{-3/2}{\sin}^{2}\theta\,\mathrm{d}\theta.

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

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19.23.4

R_{F}\left(0,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,z{\cos}^{2}% \theta\right)\,\mathrm{d}\theta=\frac{2}{\pi}\int_{0}^{\infty}R_{C}\left(y{% \cosh}^{2}t,z\right)\,\mathrm{d}t.

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function and $\int$ : integral
Referenced by:: §19.23
Permalink:: http://dlmf.nist.gov/19.23.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

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22: 19.29 Reduction of General Elliptic Integrals

§19.29 Reduction of General Elliptic Integrals

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§19.29(i) Reduction Theorems

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19.29.23

\int_{y}^{\infty}\frac{\,\mathrm{d}t}{\sqrt{(t^{2}+a^{2})(t^{2}-b^{2})}}=R_{F}% \left(y^{2}+a^{2},y^{2}-b^{2},y^{2}\right).

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/19.29.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(iii), §19.29 and Ch.19

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19.29.28

\int_{y}^{x}\frac{\,\mathrm{d}t}{\sqrt{t^{3}-a^{3}}}=4R_{F}\left(U,U-3a+2\sqrt% {3}a,U-3a-2\sqrt{3}a\right),

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $U$
Referenced by:: §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(iii), §19.29 and Ch.19

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19.29.33

(x-y)^{2}U=\left(\sqrt{x^{4}+a^{4}}+\sqrt{y^{4}+a^{4}}\right)^{2}-(x^{2}-y^{2}% )^{2}.

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Symbols:: $U$
Permalink:: http://dlmf.nist.gov/19.29.E33
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(iii), §19.29 and Ch.19

23: 19.37 Tables

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§19.37(iv) Symmetric Integrals

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24: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

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19.19.6

R_{J}\left(x,y,z,p\right)=R_{-\frac{3}{2}}\left(\tfrac{1}{2},\tfrac{1}{2},% \tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};x,y,z,p,p\right)

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.19 and Ch.19

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25: 19.30 Lengths of Plane Curves

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19.30.3

s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\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, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

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19.30.5

L(a,b)=4aE\left(k\right)=8aR_{G}\left(0,b^{2}/a^{2},1\right)=8R_{G}\left(0,a^{% 2},b^{2}\right)=8abR_{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, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $L(a,b)$ : length
Referenced by:: §19.15, §19.24(i), §19.30(i), §19.33(i), §19.9(i)
Permalink:: http://dlmf.nist.gov/19.30.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

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19.30.9

s=\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^{2}R_{D}\left(r,r+b^{2}+a^% {2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}},

r=b^{4}/y^{2}

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $I(\mathbf{m})$ : integral and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(ii), §19.30 and Ch.19

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19.30.11

s=2a^{2}\int_{0}^{r}\frac{\,\mathrm{d}t}{\sqrt{4a^{4}-t^{4}}}=\sqrt{2a^{2}}R_{% F}\left(q-1,q,q+1\right),

q=2a^{2}/r^{2}=\sec\left(2\theta\right)

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sec\NVar{z}$ : secant function, $r$ : length, $q$ and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(iii), §19.30 and Ch.19

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19.30.13

P=4\sqrt{2a^{2}}R_{F}\left(0,1,2\right)=\sqrt{2a^{2}}\times 5.24411\;51\ldots=% 4aK\left(1/\sqrt{2}\right)=a\times 7.41629\;87\dots.

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind and $P$ : length
Notes:: For more digits see OEIS Sequence A064853; see also Sloane (2003).
Permalink:: http://dlmf.nist.gov/19.30.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(iii), §19.30 and Ch.19

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26: 20.9 Relations to Other Functions

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20.9.3

R_{F}\left(\frac{{\theta_{2}}^{2}\left(z,q\right)}{{\theta_{2}}^{2}\left(0,q% \right)},\frac{{\theta_{3}}^{2}\left(z,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)},\frac{{\theta_{4}}^{2}\left(z,q\right)}{{\theta_{4}}^{2}\left(0,q% \right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{\theta_{1}\left(z,q\right)}z,

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §19.25(v), §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

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20.9.4

R_{F}\left(0,{\theta_{3}}^{4}\left(0,q\right),{\theta_{4}}^{4}\left(0,q\right)% \right)=\tfrac{1}{2}\pi,

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $q$ : nome
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

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20.9.5

\exp\left(-\frac{\pi R_{F}\left(0,k^{2},1\right)}{R_{F}\left(0,{k^{\prime}}^{2% },1\right)}\right)=q.

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $q$ : nome and $k$ : modulus
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §20.9(i), §20.9 and Ch.20

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27: 19.36 Methods of Computation

§19.36 Methods of Computation

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19.36.3

R_{F}\left(1,2,4\right)=R_{F}\left(z_{1},z_{2},z_{3}\right),

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Permalink:: http://dlmf.nist.gov/19.36.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36(i), §19.36 and Ch.19

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19.36.5

R_{F}\left(1,2,4\right)=0.68508\;58166\dots.

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.36.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36(i), §19.36 and Ch.19

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19.36.8

R_{F}\left(t_{n}^{2},t_{n}^{2}+\theta c_{n}^{2},t_{n}^{2}+\theta a_{n}^{2}\right)

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $n$ : nonnegative integer
Referenced by:: §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.36.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(ii), §19.36 and Ch.19

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19.36.11

R_{F}\left(1,2,4\right)=R_{C}\left(T^{2}+M^{2},T^{2}\right)=0.68508\;58166,

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $M$ : limit and $T$ : limit
Permalink:: http://dlmf.nist.gov/19.36.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

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28: 22.15 Inverse Functions

… ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …

29: Bibliography Z

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D. G. Zill and B. C. Carlson (1970) Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.

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External Links:: FullText, Document, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §19.21(i), §19.21(iii), §19.22(iii), §19.25(i), §19.26(i).
Encodings:: BibTeX

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30: Bibliography N

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E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.

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External Links:: ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §19.24(ii), §19.24(i), §19.9(i).
Encodings:: BibTeX

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