Generalized elliptic integrals and modular equations. (English) Zbl 0951.33012

Anderson, G. D.; Qiu, S.-L.; Vamanamurthu, M. K.; Vuorinen, M.

Pac. J. Math. 192, No. 1, 1-37 (2000).

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.

Reviewer: G.D.Anderson (East Lansing)

MSC:

33E05	Elliptic functions and integrals
33C05	Classical hypergeometric functions, \({}_2F_1\)
26D15	Inequalities for sums, series and integrals
11F03	Modular and automorphic functions

Keywords:

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