Symmetric elliptic integrals of the third kind

Authors:
D. G. Zill and B. C. Carlson

Journal:
Math. Comp. **24** (1970), 199-214

MSC:
Primary 33.19

DOI:
https://doi.org/10.1090/S0025-5718-1970-0262553-5

MathSciNet review:
0262553

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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. Bd LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. MR**0060642****[2]**B. C. Carlson,*Lauricella’s hypergeometric function 𝐹_{𝐷}*, J. Math. Anal. Appl.**7**(1963), 452–470. MR**0157017**, https://doi.org/10.1016/0022-247X(63)90067-2**[3]**B. C. Carlson,*Normal elliptic integrals of the first and second kinds*, Duke Math. J.**31**(1964), 405–419. MR**0164067****[4]**B. C. Carlson,*On computing elliptic integrals and functions*, J. Math. and Phys.**44**(1965), 36–51. MR**0175285****[5]**B. C. Carlson,*Some inequalities for hypergeometric functions*, Proc. Amer. Math. Soc.**17**(1966), 32–39. MR**0188497**, https://doi.org/10.1090/S0002-9939-1966-0188497-6**[6]**B. C. Carlson and M. D. Tobey,*A property of the hypergeometric mean value*, Proc. Amer. Math. Soc.**19**(1968), 255–262. MR**0222349**, https://doi.org/10.1090/S0002-9939-1968-0222349-X**[7]**B. C. Carlson,*A connection between elementary functions and higher transcendental functions*, SIAM J. Appl. Math.**17**(1969), 116–148. MR**0247139**, https://doi.org/10.1137/0117013**[8]**Arthur Cayley,*An elementary treatise on elliptic functions*, 2nd ed. Dover Publications, Inc., New York, 1961. MR**0124532****[9]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Higher Transcendental Functions*, McGraw-Hill, New York, 1953, Chapter 13. MR**15**, 419; MR**16**, 586.**[10]**W. J. Nellis and B. C. Carlson,*Reduction and evaluation of elliptic integrals*, Math. Comp.**20**(1966), 223–231. MR**0215497**, https://doi.org/10.1090/S0025-5718-1966-0215497-8**[11]**F. G. Tricomi,*Funzioni Ellittiche*, Zanichelli, Bologna, 1937; German transl., Akademische Verlagsgesellschaft, Geest & Portig, Leipzig, 1948. MR**10**, 532.**[12]**D. G. Zill,*Elliptic Integrals of the Third Kind*, Ph. D. Thesis, Iowa State University, Ames, Iowa, 1967.**[13]**L. M. Milne-Thomson, "Elliptic integrals," M. Abramowitz & I. Stegun (Editors),*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, Department of Commerce, Nat. Bur. Standards Appl. Math. Series, 55, U. S. Government Printing Office, Washington, D. C., 1964; 3rd printing, with corrections, 1965. MR**29**#4941; MR**31**#1400.

Retrieve articles in *Mathematics of Computation*
with MSC:
33.19

Retrieve articles in all journals with MSC: 33.19

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0262553-5

Keywords:
Elliptic integrals,
Landen transformation,
Gauss transformation,
addition theorem,
hypergeometric -functions

Article copyright:
© Copyright 1970
American Mathematical Society