Is there a holomorphic and bijective function between the open unit ball of $\mathbb{C}$ and $\mathbb{C}$?
The usual homeomorphisms $\psi(z):=\frac{z}{1+|z|}$ and his composition with the conjugate map are not holomorphic on $\mathbb{C}$.
2026-03-26 11:00:14.1774522814
Is there a holomorphic and bijective function of the unit disc onto the complex plane?
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No, because its inverse would be a holomorphic map from $\mathbb C$ to the open unit disk on $\mathbb C$. Therefore, it would be bounded and then, by Liouville's theorem, it would be a constant map.