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[parent] failure of Hartogs' theorem in one dimension (Example)

It is instructive to see an example where Hartogs' theorem fails in one dimension. Take $ U = {\mathbb{C}}$ and let $ K = \{0\}.$ The function $ \frac{1}{z}$ is holomorphic in $ U \setminus K,$ but cannot be extended to $ U.$

To understand the example and failure of the theorem it is important to understand the proof. In the proof, the way we construct an extension is that we start with a function holomorphic in $ U \setminus K,$ modify it in a neighbourhood of $ K$ to be zero, hence extending as a smooth function through $ K.$ Then we solve the $ \bar{\partial}$ operator inhomogeneous equation $ \bar{\partial}\psi = g$ to “correct” our extension to be holomorphic. The key point is that $ g$ has compact support allowing us to solve the equation and find a $ \psi$ with compact support. This fails in dimension 1. While we always get a solution $ \psi,$ the solution can never have compact support. Hence, if we tried the proof with $ \frac{1}{z},$ the new function we obtain in the proof does not agree with $ \frac{1}{z}$ on any open set and hence is not an extension.



"failure of Hartogs' theorem in one dimension" is owned by jirka.
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Cross-references: open set, solution, support, compact, point, equation, inhomogeneous, smooth function, neighbourhood, extension, proof, holomorphic, function, dimension, Hartogs theorem

This is version 1 of failure of Hartogs' theorem in one dimension, born on 2008-02-06.
Object id is 10242, canonical name is FailureOfHartogsTheoremInOneDimension.
Accessed 219 times total.

Classification:
AMS MSC32H02 (Several complex variables and analytic spaces :: Holomorphic mappings and correspondences :: Holomorphic mappings, embeddings and related questions)

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