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Zbl 0951.33012
Anderson, G.D.; Qiu, S.-L.; Vamanamurthu, M.K.; Vuorinen, M.
Generalized elliptic integrals and modular equations. (English)
[J] Pac. J. Math. 192, No.1, 1-37 (2000). ISSN 0030-8730

Authors' summary: In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan's modular equations and approximations to $\pi$. The authors study the monotonicity and convexity properties of these quantities and obtain sharp inequalities for them.
[G.D.Anderson (East Lansing)]

MSC 2000:: *33E05 Elliptic functions and integrals
33C05 Classical hypergeometric functions
26D15 Inequalities for sums, series and integrals of real functions
11F03 Modular and automorphic functions

Keywords: zero-balanced hypergeometric functions; modular equations; signature; degree; conformal; modulus

Cited in: Zbl 1101.33002 Zbl 1098.33015 Zbl 1110.11300 Zbl 0983.33002

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