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9 votes
1 answer
3k views

If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
O.R.'s user avatar
  • 807
4 votes
1 answer
447 views

Hartogs's extension theorem

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$, and let $f:H\to \mathbb{C}^n$ be a holomorphic injective map. Then we know that $f$ extends holomorphically to the polydisc $P$, i.e. there ...
felipeuni's user avatar
  • 155
3 votes
0 answers
848 views

Does a bounded convex domain has one smooth boundary point?

In the study of analysis and geometry of a bounded domain, its boundary regularity is important. For example, it is known that a bounded convex domain has Lipschitz bounday. This implies that a ...
Entaou's user avatar
  • 285
0 votes
0 answers
49 views

pseudo inverse of a holomorphic multivariate injective map

Let $f:{\mathbb C}^n \rightarrow {\mathbb C}^N$, $N > n$, be holomorphic and injective on an open ball $B_n \subset {\mathbb C}^n$ such that the Jacobian matrices have full column rank at each ...
gil's user avatar
  • 265