Difference between revisions of "Domain, double-circled"
From Encyclopedia of Mathematics
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| − | A domain | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$z_j=\{a_j+(z_j^0-a_j)e^{i\phi_j}\},\quad0\leq\phi_j\leq2\pi,\quad j=1,2,$$ | |
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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]], is defined in a similar manner. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hörmander, | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4</TD></TR> | ||
| + | </table> | ||
Latest revision as of 06:05, 9 July 2024
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.
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=11787
Domain, double-circled. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Domain,_double-circled&oldid=11787
This article was adapted from an original article by M. Shirinbekov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article