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 Mat. Fiz. Anal. Geom., 1996, Volume 3, Number 3/4, Pages 290–307 (Mi jmag498)

The characterization of conformal maps of the upper halfplane on a “comb” type domain

A. V. Kesarev

Kharkiv State University

Abstract: The domain $ż\in\mathbf C: -\infty\leq a<\operatorname{Re}z<b\leq+\infty,\operatorname{Im}z>0\}\setminus\{\cup[x_k,x_k+iy_k]\}$ is called a “comb” type domain. For each closed set $E$ on the real axis there exists the unique conformal map of the upper halfplane onto a certain “comb” type domain of mapping the set $E$ on the interval $(a,b)$. If $a=-\infty$ and $b=+\infty$, then the set $E$ is referred to the type $(A)$. If either $a=-\infty$, $b<+\infty$, or $a>-\infty$, $b=+\infty$, then $E$ is referred to the type $(B)$. If both $a$ and $b$ are finite, then $E$ is referred to the type $(C)$. Conditions for a set $E$ to be referred to the type $(A)$, $(B)$ or $(C)$ are given.

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Citation: A. V. Kesarev, “The characterization of conformal maps of the upper halfplane on a “comb” type domain”, Mat. Fiz. Anal. Geom., 3:3/4 (1996), 290–307

Citation in format AMSBIB
\Bibitem{Kes96} \by A.~V.~Kesarev \paper The characterization of conformal maps of the upper halfplane on a comb'' type domain \jour Mat. Fiz. Anal. Geom. \yr 1996 \vol 3 \issue 3/4 \pages 290--307 \mathnet{http://mi.mathnet.ru/jmag498} 

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• http://mi.mathnet.ru/eng/jmag/v3/i3/p290

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This publication is cited in the following articles:
1. Eremenko A., Yuditskii P., “Comb Functions”, Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, Contemporary Mathematics, 578, eds. Arvesu J., Lagomasino G., Amer Mathematical Soc, 2011, 99+