# Hartogs domain

semi-circular domain, with symmetry plane $\{z_n=a_n\}$

A domain in the space of $n$ complex variables which, for each point $z=(z_1,\dots,z_{n-1},z_n)\equiv('z,z_n)$, contains the circle

$$\left\{('z,a_n+e^{i\theta}(z_n-a_n)):0\leq\theta<2\pi\right\}.$$

Named after F. Hartogs. A Hartogs domain is called complete if for each point $('z,z_n)$ it contains the disc

$$\{('z,a_n+\lambda(z_n-a_n)):|\lambda|\leq1\}.$$

A Hartogs domain with symmetry plane $\{z_n=0\}$ can conveniently be represented by a Hartogs diagram, viz., by the image of the Hartogs domain under the mapping $('z,z_n)\to('z,|z_n|)$.

#### References

 [1] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian) [2] S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948)