Singular values, Ramanujan modular equations, and Landen transformations
Vuorinen, M.
Studia Mathematica, Tome 119 (1996), p. 221-230 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

A new connection between geometric function theory and number theory is derived from Ramanujan’s work on modular equations. This connection involves the function φK(r) recurrent in the theory of plane quasiconformal maps. Ramanujan’s modular identities yield numerous new functional identities for φ1/p(r) for various primes p.

Publié le : 1996-01-01
EUDML-ID : urn:eudml:doc:216353 
@article{bwmeta1.element.bwnjournal-article-smv121i3p221bwm,
     author = {M. Vuorinen},
     title = {Singular values, Ramanujan modular equations, and Landen transformations},
     journal = {Studia Mathematica},
     volume = {119},
     year = {1996},
     pages = {221-230},
     zbl = {0872.30010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv121i3p221bwm}
}

Vuorinen, M. Singular values, Ramanujan modular equations, and Landen transformations. Studia Mathematica, Tome 119 (1996) pp. 221-230. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv121i3p221bwm/

[00000] [AB] G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly 95 (1988), 585-608. | Zbl 0665.26007

[00001] [AV] G. D. Anderson and M. K. Vamanamurthy, Some properties of quasiconformal distortion functions, New Zealand J. Math. 24 (1995), 1-15. | Zbl 0857.30019

[00002] [AVV1] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Functional inequalities for complete elliptic integrals and their ratios, SIAM J. Math. Anal. 21 (1990), 536-549. | Zbl 0692.33001

[00003] [AVV2] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for plane quasiconformal mappings, in: The Mathematical Heritage of Wilhelm Magnus - Groups, Geometry and Special Functions, W. Abikoff, J. S. Birman and K. Kuiken (eds.), Contemp. Math. 169, Amer. Math. Soc., 1994, 1-27.

[00004] [B] B. C. Berndt, Ramanujan Notebooks, Part III, Springer, Berlin, 1991.

[00005] [BBG] 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. | Zbl 0843.33012

[00006] [BC] B. C. Berndt and H. H. Chan, Ramanujan's explicit values for the classical theta-functions, Mathematika 42 (1995), 278-294. | Zbl 0862.33016

[00007] [BB] J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, New York, 1987.

[00008] [FK] H. M. Farkas and Y. Kopeliovich, New theta constant identities II, Proc. Amer. Math. Soc. 123 (1995), 1009-1020. | Zbl 0826.11018

[00009] [H] G. H. Hardy, Ramanujan, Chelsea, New York, 1940.

[00010] [J] C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, 1829, Jacobi's Gesammelte Werke, Vol. 1, Berlin, 1881-1891.

[00011] [KZ] W. Kawa and J. Zając, Dynamical approximation of the distortion function ΦK, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 20 (1995), 39-48. | Zbl 0876.30016

[00012] [L] D. F. Lawden, Elliptic Functions and Applications, Springer, New York, 1989. | Zbl 0689.33001

[00013] [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973. | Zbl 0267.30016

[00014] [P1] D. Partyka, Approximation of the Hersch-Pfluger distortion function. Applications, Ann. Univ. Mariae Curie-Skłodowska Sect. A 45 (1991), 99-111.

[00015] [P2] D. Partyka, Approximation of the Hersch-Pfluger distortion function, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), 343-354. | Zbl 0793.30013

[00016] [P3] D. Partyka, The maximal value of the function [0,1]∋r→ΦK2(√r)-r, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 20 (1995), 49-55. | Zbl 0876.30015

[00017] [QVV] S.-L. Qiu, M. K. Vamanamurthy and M. Vuorinen, Bounds for quasiconformal distortion functions, J. Math. Anal. Appl., to appear. | Zbl 0871.30019

[00018] [SC] A. Selberg and S. Chowla, On Epstein's zeta-function, J. Reine Angew. Math. 227 (1967), 87-110. | Zbl 0166.05204

[00019] [S1] L.-C. Shen, On some cubic modular identities, Proc. Amer. Math. Soc. 119 (1993), 203-208. | Zbl 0781.33008

[00020] [S2] L.-C. Shen, On some modular equations of degree 5, ibid. 123 (1995), 1521-1526. | Zbl 0826.33011

[00021] [VV] M. K. Vamanamurthy and M. Vuorinen, Functional inequalities, Jacobi products and quasiconformal maps, Illinois J. Math. 38 (1994), 394-419. | Zbl 0799.30011