View source for Domain, double-circled

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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

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

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====
<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>