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

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.


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