Domain, double-circled

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