Moduli of quadrilaterals and quasiconformal reflection

Semen Nasyrov, Toshiyuki Sugawa, Matti Vuorinen

Research output: Contribution to journalArticlepeer-review


We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. Then we study the special case of an isosceles trapezoidal polygon L and obtain two-sided estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.

Original languageEnglish
Article number127092
JournalJournal of Mathematical Analysis and Applications
Issue number2
Publication statusPublished - 2023 Aug 15


  • Appell hypergeometric functions
  • Elliptic integrals
  • Extremal length
  • Gauss hypergeometric functions
  • Modulus of a quadrilateral
  • Quasiconformal reflection

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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