Hartogs's extension theorem

In mathematics, especially several complex variables, Hartogos extension theorem states:

If $f$ is holomorphic in $G/K$, then $f$ can be extended to a unique holomorphic to $G$, where $G\subset \mathbf {C} ^{n}$ is an open subset of $\mathbf {C} ^{n}$ (with $n\geq 2$) and K is a compact subset of G such that$G/K$ is connected.

The theorem does not hold when $n=1$. It thus constitutes one of elementary phenomena that is unique to several complex variables.

Usual proofs rely on either Bochner-Martinelli-Koppelman formula or on the solution of the inhomogeneous Cauchy-Riemann equation with compact support. The latter approach is due to Ehrenpreis.