The Hartogs figure H(r,s) and its bidisc hull

Fix real numbers $0<r,s<1$. The Hartogs figure $H(r,s)\subseteq {\mathbb{C}}^2$ is

$H(r,s):=\{(z_1,z_2):\mathrm{\mid }{z}_1\mathrm{\mid }<1,\mathrm{\mid }{z}_2\mathrm{\mid }<s\}\cup \{(z_1,z_2):r<\mathrm{\mid }{z}_1\mathrm{\mid }<1,\mathrm{\mid }{z}_2\mathrm{\mid }<1\}\mathrm{.}$

Its bidisc hull is the full unit bidisc

$\widehat{H}(r,s):=\{(z_1,z_2):\mathrm{\mid }{z}_1\mathrm{\mid }<1,\mathrm{\mid }{z}_2\mathrm{\mid }<1\}={\mathrm{\Delta }}_1(0)\times {\mathrm{\Delta }}_1(0)\mathrm{.}$

The first piece is the thin cylinder over the $z_1$-disc, and the second piece is the thick outer shell in the $z_1$-variable with full $z_2$-disc available. The missing core is $\{(z_1,z_2):\mathrm{\mid }{z}_1\mathrm{\mid }\le r,s\le \mathrm{\mid }{z}_2\mathrm{\mid }<1\}$, and the Hartogs phenomenon is that holomorphic functions on $H(r,s)$ do not feel that missing core.

Translated and rescaled versions are obtained by applying affine complex coordinate changes to the bidisc description above; the later shell-extension lemma uses exactly that coordinate model.