Symmetric elliptic integrals of the third kind

Zill, D. G.; Carlson, B. C.

doi:10.1090/S0025-5718-1970-0262553-5

Symmetric elliptic integrals of the third kind
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by D. G. Zill and B. C. Carlson PDF

Math. Comp. 24 (1970), 199-214 Request permission

Abstract:

Legendre’s incomplete elliptic integral of the third kind can be replaced by an integral which possesses permutation symmetry instead of a set of linear transformations. Two such symmetric integrals are discussed, and direct proofs are given of properties corresponding to the following parts of the Legendre theory: change of parameter, Landen and Gauss transformations, interchange of argument and parameter, relation of the complete integral to integrals of the first and second kinds, and addition theorem. The theory of the symmetric integrals offers gains in simplicity and unity, as well as some new generalizations and some inequalities.

References

Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Band LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. MR 0060642
B. C. Carlson, Lauricella’s hypergeometric function $F_{D}$, J. Math. Anal. Appl. 7 (1963), 452–470. MR 157017, DOI 10.1016/0022-247X(63)90067-2
B. C. Carlson, Normal elliptic integrals of the first and second kinds, Duke Math. J. 31 (1964), 405–419. MR 164067
B. C. Carlson, On computing elliptic integrals and functions, J. Math. and Phys. 44 (1965), 36–51. MR 175285
B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 17 (1966), 32–39. MR 188497, DOI 10.1090/S0002-9939-1966-0188497-6
B. C. Carlson and M. D. Tobey, A property of the hypergeometric mean value, Proc. Amer. Math. Soc. 19 (1968), 255–262. MR 222349, DOI 10.1090/S0002-9939-1968-0222349-X
B. C. Carlson, A connection between elementary functions and higher transcendental functions, SIAM J. Appl. Math. 17 (1969), 116–148. MR 247139, DOI 10.1137/0117013
Arthur Cayley, An elementary treatise on elliptic functions, Dover Publications, Inc., New York, 1961. 2nd ed. MR 0124532

Higher Transcendental Functions

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W. J. Nellis and B. C. Carlson, Reduction and evaluation of elliptic integrals, Math. Comp. 20 (1966), 223–231. MR 215497, DOI 10.1090/S0025-5718-1966-0215497-8

Funzioni Ellittiche

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Elliptic Integrals of the Third Kind

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

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