Generalized elliptic integrals and modular equations

Pacific Journal of Mathematics

Volumn 192, Issue 1, 2000, Pages 1-37

  Anderson, G D ^a   Qiu, S L ^b   Vamanamurthy, M K ^c   Vuorinen, M ^d  

^a MICHIGAN STATE UNIVERSITY   (United States)

^b HANGZHOU DIANZI UNIVERSITY   (China)

^c UNIVERSITY OF AUCKLAND   (New Zealand)

^d NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001873330     PISSN: 00308730     EISSN: None     Source Type: Journal    
DOI: 10.2140/pjm.2000.192.1     Document Type: Article

References (31)

1. M. Abramowitz and I.A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965.
2. L.V. Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.
3. G.D. Anderson, R.W. Barnard, K.C. Richards, M.K. Vamanamurthy and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc., 347 (1995), 1713-1723.
4. G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.
5. R. Askey, Ramanujan and hypergeometric and basic hypergeometric series, Ramanujan Internat. Symposium on Analysis, December 26-28, 1987, ed. by N.K. Thakare, 1-83, Pune, India, Russian Math. Surveys, 451 (1990), 37-86.
6. R. Balasubramanian, S. Ponnusamy and M. Vuorinen, Functional inequalities for the quotients of hypergeometric functions, J. Math. Anal. Appl., 218 (1998), 256-268.
7. Bateman Manuscript Project (ed. A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi), Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
8. B.C. Berndt, Ramanujan's Notebooks, Vol. I, Springer-Verlag, Berlin-Heidelberg-New York, 1985.
9. _, Ramanujan's Notebooks, Vol. II, Springer-Verlag, Berlin-Heidelberg-New York, 1989.
10. _, Ramanujan's Notebooks, Vol. III, Springer-Verlag, Berlin-Heidelberg-New York, 1991.
11. B.C. Berndt, S. Bhargava and F.G. Garvan, Ramanujan's theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc., 347 (1995), 4163-4244.
12. J.M. Borwein and P.B. Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987.
13. _, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), 691-701.
14. F. Bowman, Introduction to Elliptic Functions with Applications, Dover, New York, 1961.
15. P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, 2nd ed., Die Grundlehren Math. Wiss, 67, Springer-Verlag, Berlin-Göttingen-Heidelberg-New York, 1971.
16. I. Chavel, Riemannian Geometry - A Modern Introduction, Cambridge Tracts in Math., 108, Cambridge Univ. Press, 1993.
17. A.L. Dixon, Generalizations of Legendre's formula KE′ - (K - E)K′ = 1/2π, Proc. London Math. Soc, 3(2) (1905), 206-224.
18. P.L. Duren, The Legendre relation for elliptic integrals, in Paul Halmos: Celebrating 50 Years of Mathematics, ed. by J.H. Ewing and F.W. Gehring, Springer-Verlag, New York, (1991), 305-315
19. E.B. Elliott, A formula including Legendre's EK′ + KE′ - KK′ = 1/2π, Messenger of Math., 33 (1904), 31-40.
20. F.G. Garvan, Ramanujan's theories of elliptic functions to alternative bases - a symbolic excursion, J. Symbolic Comput., 20 (1995), 517-536.
21. O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Die Grundlehren der Math. Wiss, 126, Springer-Verlag, New York-Berlin, 1973.
22. A.I. Markushevich, Theory of Functions of a Complex Variable, Vol. II, Prentice-Hall, Englewood Cliffs, NJ, 1965.
23. G.J. Martin, The distortion theorem for quasiconformal mappings, Schottky's theorem and holomorphic motions, Proc. Amer. Math. Soc., 125 (1997), 1095-1103.
24. W. Miller, Jr., Symmetry and separation of variables, With a foreword by Richard Askey; Encyclopedia of Mathematics and its Applications, Vol. 4, Reading, Massachusetts, Addison-Wesley Publishing Company, XXX (1977).
25. S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44 (1997), 278-301.
26. S.-L. Qiu and M. Vuorinen, Landen inequalities for hypergeometric functions, Nagoya Math. J., 154 (1999), 31-56.
27. _, Duplication inequalities for the ratios of hypergeometric functions, Forum Math., 11 (1999), to appear.
28. _, Infinite products and the normalized quotients of hypergeometric functions, SIAM J. Math. Anal., 30 (1999), 1057-1075.
29. E.D. Rainville, Special Functions, MacMillan, New York, 1960.
30. V.S. Varadarajan, Linear meromorphic differential equations: A modern point of view, Bull. Amer. Math. Soc., 33 (1996), 1-42.
31. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, London, 1958.