Biholomorphic map $f: \Bbb C \rightarrow \Bbb C$ with $f(z_1) = z_2$

Question

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.

6

Why not $f(z)=z_2+(z-z_1)$? – Clayton Jun 28 '18 at 19:36

score 3 · Accepted Answer · answered Jun 28 '18 at 19:50

3

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 $a \in \Bbb C \setminus \{ 0 \}$.

answered Jun 28 '18 at 19:50

Martin R

113,040

Biholomorphic map $f: \Bbb C \rightarrow \Bbb C$ with $f(z_1) = z_2$

1 Answers1