Vol. 192, No. 1, 2000

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Generalized elliptic integrals and modular equations

G.D. Anderson, S.-L. Qiu, M.K. Vamanamurthy and M. Vuorinen

Vol. 192 (2000), No. 1, 1–37
Abstract

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 π. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them.

Milestones
Received: 23 April 1998
Published: 1 January 2000
Authors
G.D. Anderson
Michigan State University
East Lansing, MI 48824
S.-L. Qiu
Hangzhou Institute of Electronics Engineering
Hangzhou 310037
P.R. China
M.K. Vamanamurthy
University of Auckland
Auckland
New Zealand
M. Vuorinen
University of Helsinki
FIN-00014
Finland