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Difference between revisions of "Domain, double-circled"

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A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337701.png" /> in the two-dimensional complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337702.png" /> having the following property: There is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337703.png" /> such that, with each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337704.png" />, all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337705.png" /> with coordinates
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A domain $D$ in the two-dimensional complex space $\mathbf C^2$ having the following property: There is a point $(a_1,a_2)$ such that, with each point $(z_1^0,z_2^0)$, all points $(z_1,z_2)$ with coordinates
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337706.png" /></td> </tr></table>
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$$z_j=\{a_j+(z_j^0-a_j)e^{i\phi_j}\},\quad0\leq\phi_j\leq2\pi,\quad j=1,2,$$
  
belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337707.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d0337708.png" /> is said to be the centre of the double-circled domain. If a double-circled domain contains its own centre, it is said to be complete; if it does not, it is called incomplete. Examples of complete double-circled domains are a sphere or a bicylinder; examples of an incomplete double-circled domain include the Cartesian product of annuli. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033770/d03377010.png" />-circled domain, or a [[Reinhardt domain|Reinhardt domain]], is defined in a similar manner.
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belong to $D$. The point $(a_1,a_2)$ is said to be the centre of the double-circled domain. If a double-circled domain contains its own centre, it is said to be complete; if it does not, it is called incomplete. Examples of complete double-circled domains are a sphere or a bicylinder; examples of an incomplete double-circled domain include the Cartesian product of annuli. An $n$-circled domain, or a [[Reinhardt domain|Reinhardt domain]], is defined in a similar manner.
  
  

Revision as of 14:23, 30 July 2014

A domain $D$ in the two-dimensional complex space $\mathbf C^2$ having the following property: There is a point $(a_1,a_2)$ such that, with each point $(z_1^0,z_2^0)$, all points $(z_1,z_2)$ with coordinates

$$z_j=\{a_j+(z_j^0-a_j)e^{i\phi_j}\},\quad0\leq\phi_j\leq2\pi,\quad j=1,2,$$

belong to $D$. The point $(a_1,a_2)$ is said to be the centre of the double-circled domain. If a double-circled domain contains its own centre, it is said to be complete; if it does not, it is called incomplete. Examples of complete double-circled domains are a sphere or a bicylinder; examples of an incomplete double-circled domain include the Cartesian product of annuli. An $n$-circled domain, or a Reinhardt domain, is defined in a similar manner.


Comments

References

[a1] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4
How to Cite This Entry:
Domain, double-circled. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Domain,_double-circled&oldid=32565
This article was adapted from an original article by M. Shirinbekov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article