Computational Methods and Function Theory 9 (2009), No. 1, 75--109
Copyright Heldermann Verlag 2009
Ville Heikkala
ville.heikkala@ssh.com , SSH Communications Security Corp., Valimotie 17, FIN-00380 Helsinki, Finland.
Mavina K. Vamanamurthy
vamanamu@math.auckland.ac.nz , The University of Auckland, Department of Mathematics, P B 92019, Auckland, New Zealand.
Matti Vuorinen
vuorinen@utu.fi , University of Turku, Department of Mathematics, FIN-20014 University of Turku, Finland.
Jacobi's elliptic integrals and elliptic functions arise naturally from the Schwarz-Christoffel conformal transformation of the upper half plane onto a rectangle. In this paper we study generalized elliptic integrals which arise from the analogous mapping of the upper half plane onto a quadrilateral and obtain sharp monotonicity and convexity properties for certain combinations of these integrals, thus generalizing analogous well-known results for classical conformal capacity and quasiconformal distortion functions. An algorithm for the computation of the modulus of the quadrilateral is given.
Keywords: Generalized elliptic integrals, modulus of a quadrilateral.
MSC 2000: Primary 33B15, 33C05; Secondary 30C62.
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