Bochner's tube theorem

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

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

A classic reference is ^[1] (Theorem 9). See also ^[2]^[3] for other proofs.

^ Bochner, S.; Martin, W.T. (1948). Several Complex Variables. Princeton mathematical series. Princeton University Press. ISBN 978-0-598-34865-4.
^ Hounie, J. (2009). "A Proof of Bochner's Tube Theorem". Proceedings of the American Mathematical Society. 137 (12). American Mathematical Society: 4203–4207. doi:10.1090/S0002-9939-09-10057-6. JSTOR 40590656.
^ Noguchi, Junjiro (2020). "A brief proof of Bochner's tube theorem and a generalized tube". arXiv:2007.04597 [math.CV].

[Bochner_Martin_Haar_1948_p.-1] Bochner, S.; Martin, W.T. (1948). Several Complex Variables. Princeton mathematical series. Princeton University Press. ISBN 978-0-598-34865-4.

[Hounie_2009-2] Hounie, J. (2009). "A Proof of Bochner's Tube Theorem". Proceedings of the American Mathematical Society. 137 (12). American Mathematical Society: 4203–4207. doi:10.1090/S0002-9939-09-10057-6. JSTOR 40590656.

[3] Noguchi, Junjiro (2020). "A brief proof of Bochner's tube theorem and a generalized tube". arXiv:2007.04597 [math.CV].

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