Difference between revisions of "Attainable boundary arc"
From Encyclopedia of Mathematics
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| − | ''of a domain | + | ''of a domain $G$ in the $z$-plane'' |
| − | A Jordan arc forming part of the boundary of | + | A Jordan arc forming part of the boundary of $G$ and at the same time forming part of the boundary of some Jordan domain $g \subset G$. Each point on an attainable boundary arc is an attainable (from the inside of $g$) boundary point of $G$ (cf. [[Attainable boundary point]]). A conformal mapping of a simply-connected domain $G$ onto the unit disc $D = \{ z : |z| < 1 \}$ can be continuously extended to the non-terminal points of an attainable boundary arc to a homeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$. |
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
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| + | {{TEX|done}} | ||
Latest revision as of 19:25, 13 December 2015
of a domain $G$ in the $z$-plane
A Jordan arc forming part of the boundary of $G$ and at the same time forming part of the boundary of some Jordan domain $g \subset G$. Each point on an attainable boundary arc is an attainable (from the inside of $g$) boundary point of $G$ (cf. Attainable boundary point). A conformal mapping of a simply-connected domain $G$ onto the unit disc $D = \{ z : |z| < 1 \}$ can be continuously extended to the non-terminal points of an attainable boundary arc to a homeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$.
Comments
The standard Western terminology is accessible boundary arc and accessible boundary point, see e.g. [a1].
References
| [a1] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9 |
| [a2] | A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 2 (Translated from Russian) |
How to Cite This Entry:
Attainable boundary arc. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attainable_boundary_arc&oldid=36924
Attainable boundary arc. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attainable_boundary_arc&oldid=36924
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article