Abstract
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.
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The first author is supported by the Magnus Ehrnrooth fund of the Finnish Academy of Science and Letters.
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Heikkala, V., Vamanamurthy, M.K. & Vuorinen, M. Generalized Elliptic Integrals. Comput. Methods Funct. Theory 9, 75–109 (2009). https://doi.org/10.1007/BF03321716
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DOI: https://doi.org/10.1007/BF03321716