Let $z_1, z_2 \in \Bbb C$. I want to find a biholomorphic map $f: \Bbb C \rightarrow \Bbb C$ such that $f(z_1) = z_2$, but I do not know how such a map could look like.
2026-03-25 19:05:14.1774465514
Biholomorphic map $f: \Bbb C \rightarrow \Bbb C$ with $f(z_1) = z_2$
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A biholomorphic map from $\Bbb C$ onto itself is necessarily linear (see e.g. entire 1-1 function), which means that all solutions are given by $$ f(z) = z_2 + a(z - z_1) $$ for some $a \in \Bbb C \setminus \{ 0 \}$.