Bochner's tube theorem

In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in ${\displaystyle \mathbb {C} ^{n}}$ can be extended to the convex hull of this domain.

Theorem Let ${\displaystyle \omega \subset \mathbb {R} ^{n}}$ be a connected open set. Then every function ${\displaystyle f(z)}$ holomorphic on the tube domain ${\displaystyle \Omega =\omega +i\mathbb {R} ^{n}}$ can be extended to a function holomorphic on the convex hull ${\displaystyle \operatorname {ch} (\Omega )}$.

A classic reference is Bochner & Martin (1948) (Theorem 9). See also Hounie (2009) and Noguchi (2020) for other proofs.