08/01/2022, 11:15 PM
I'm confused, wasn't this shown to be non-analytic?
I believe this will just construct a smooth solution on \(\mathbb{R}\).
Additionally, the semi-group homomorphism is only unique in the neighborhood of a fixed point. There are plenty of semi-group homomorphisms that are holomorphic, they just can't be holomorphic at the fixed point (At a neighborhood \( p \in \mathcal{N} = \{|z-p| < \delta\}\).
There are countably infinite semi-group homomorphisms. The trouble is, there's only one that is holomorphic about a fixed point. That's the uniqueness statement.
I believe this will just construct a smooth solution on \(\mathbb{R}\).
Additionally, the semi-group homomorphism is only unique in the neighborhood of a fixed point. There are plenty of semi-group homomorphisms that are holomorphic, they just can't be holomorphic at the fixed point (At a neighborhood \( p \in \mathcal{N} = \{|z-p| < \delta\}\).
There are countably infinite semi-group homomorphisms. The trouble is, there's only one that is holomorphic about a fixed point. That's the uniqueness statement.