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For compact $K$ in a plane domain $\Omega$, its holomorphic convex hull isIt is $K$ together with the components of $\hat{\mathbb{C}}\setminus K$ lying entirely in $\Omega$. Inclusion of those components follows from the maximum modulus principle. For exclusion, suppose $w$ lies in a complementary component meeting $\hat{\mathbb{C}}\setminus \Omega$. Choose an allowed pole there. The open-and-closed resolvent argument in the proof of Runge theorem approximates $(z-w{)}^{-1}$ uniformly on $K$ by functions $h$ holomorphic on $\Omega$. Then $1-(z-w)h(z)$ has value one at $w$ and arbitrarily small modulus on $K$, separating $w$ from the hull. For $\Omega =\mathbb{C}$, this is the polynomial hull in one complex variable.