symmetries

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31: 19.25 Relations to Other Functions

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19.25.16

\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2% },c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2% })}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})% \right),

\omega^{2}=k^{2}/\alpha^{2}

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.25(i)
Permalink:: http://dlmf.nist.gov/19.25.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(i), §19.25 and Ch.19

►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). … ►(

{F_{1}}

and

F_{D}

are equivalent to the

R

-function of 3 and

n

variables, respectively, but lack full symmetry.) …

32: 19.30 Lengths of Plane Curves

… ►showing the symmetry in

a

and

b

. …

33: Bibliography

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D. Atkinson and P. W. Johnson (1988) Chiral-symmetry breaking in QCD. I. The infrared domain. Phys. Rev. D (3) 37 (8), pp. 2290–2295.

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External Links:: ISSN 0556-2821, Document, MathReview
Referenced by:: §15.18.
Encodings:: BibTeX

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34: Bibliography G

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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