Runge domain

Remarks

Example: the unit polydisc $\Delta^n$. Every holomorphic $f$ on $\Delta^n$ is the sum of its Taylor series at $0$, and this series converges uniformly on each compact subset of $\Delta^n$; its partial sums are polynomials. More generally every convex open set is a Runge domain, by the Oka–Weil theorem applied to the convex hull of $K$, which is compact and polynomially convex.

Non-example: $D = \set{z \in \C^n : 1 < \abs{z_1} < 2}$ and $f(z) = 1/z_1$. On the compact circle $K = \set{\abs{z_1} = 3/2, \ z_2 = \cdots = z_n = 0}$, every polynomial $p$ satisfies $\int_{\abs{\zeta} = 3/2} p(\zeta, 0, \dots, 0) \, d\zeta = 0$, while the same integral of $f$ is $2\pi i$. So $f$ is not a uniform limit of polynomials on $K$.

For $n = 1$, Runge's theorem gives a complete answer. An open set $D \subseteq \C$ is a Runge domain exactly when every connected component of $D$ is simply connected.