
Conformal mapping of halfplane onto circual numerable polygon with double symmetry
I. A. Kolesnikov^{} ^{} Tomsk State University, Faculty of Mechanics and Mathematics
Abstract:
Recently conformal mapping of the upper halfplane onto simply connected domains of the halfplane type with the symmetry of transfer along the real axis by $2\pi$, with a boundary consisting of circular arcs, straight line segments and rays have been used in mathematical physics. In the paper it is proved that the conformal mapping of the upper halfplane onto such domain that has the additional property of symmetry with respect to the vertical straight $\omega=\pi+i\upsilon, \upsilon\in \mathbb{R}$ is a solution of a differential equation of the third order of ChristoffelSchwarz equation type for circular polygons. The received equation depends on the values of the angles at the finite number of vertices, their counter images, the accessory parameters. The proof is based on the RiemannSchwarz principle of symmetry and the ChristoffelSchwarz formula for circular polygons. The system of two linear algebraic equations for the accessory parameters has been written. For mapping onto the specific circular numerablepolygon with double symmetry, the diffenerential equation, equivalent to the Fuchs class equation with three singular points, has been reduced to the Gauss equation. The map is represented in terms of hypergeometric integrals.
Keywords:
circular numerable polygon; conformal mapping; symmetry of transfer; Schwartz derivative; Gauss equation
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UDC:
517.542
MSC: 30C20 Received: 05.07.2013
Citation:
I. A. Kolesnikov, “Conformal mapping of halfplane onto circual numerable polygon with double symmetry”, Probl. Anal. Issues Anal., 2(20):2 (2013), 59–67
Citation in format AMSBIB
\Bibitem{Kol13}
\by I.~A.~Kolesnikov
\paper Conformal mapping of halfplane onto circual numerable polygon with double symmetry
\jour Probl. Anal. Issues Anal.
\yr 2013
\vol 2(20)
\issue 2
\pages 5967
\mathnet{http://mi.mathnet.ru/pa7}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3039586}
\zmath{https://zbmath.org/?q=an:1294.30015}
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