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I assume the following is a standard consideration and question, but I don't know how to prove it:

There is a trivial one-to-one correspondence between affine transformations $f = (f_x,f_y)$ of the plane $\mathbb{R}^2$:

$$\begin{bmatrix} f_x \\ f_y \\ \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\begin{bmatrix} x \\ y \\ \end{bmatrix} + \begin{bmatrix} e \\ f \\ \end{bmatrix} $$

with $ad - bc \neq 0$ and complex functions $g(u + iv) = g_u(u,v) + ig_v(u,v)$ with

$$\begin{bmatrix} g_u \\ g_v \\ \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\begin{bmatrix} u \\ v \\ \end{bmatrix} + \begin{bmatrix} e \\ f \\ \end{bmatrix} $$

Some such functions are entire functions, e.g. the identity function

$$\begin{bmatrix} g_u \\ g_v \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}\begin{bmatrix} u \\ v \\ \end{bmatrix} $$

others are not, e.g. the complex conjugate

$$\begin{bmatrix} g_u \\ g_v \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}\begin{bmatrix} u \\ v \\ \end{bmatrix} $$

which corresponds to reflection at the real axis.

I guess that the only affine transformations that correspond to entire complex functions are compositions of (uniform) scalings, rotations, and translations. Especially reflections don't qualify. Of non-uniform scalings and shearings (= non-uniform translations) I doubt it.

The qualifiying transformations would simply correspond to functions

$$g(z) = z_1\cdot z + z_0$$

with $z_0 = e + if$ and

$$ \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} = r\begin{bmatrix} \cos(\varphi) & \sin(\varphi) \\ -\sin(\varphi) & \cos(\varphi) \\ \end{bmatrix}$$

for $z_1 = re^{i\varphi}$.

My questions are:

  1. Is this correct?

  2. How to prove it?

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Yes, it is correct. I suppose that you know how to prove that functions of the type $z\mapsto z_1\times z+z_0$ are entire. Now, suppose that $a,b,c,d,e,f\in\mathbb R$ and that you consider the map$$\begin{array}{rccc}F\colon&\mathbb C&\longrightarrow&\mathbb C\\&x+yi&\mapsto&ax+by+e+(cx+dy+f)i.\end{array}$$Is it entire? It is clear that it is entire if and only if the map$$\begin{array}{ccc}\mathbb C&\longrightarrow&\mathbb C\\x+yi&\mapsto&ax+by+(cx+dy)i\end{array}$$is entire. But if it is entire, then it is holomorphic and the Cauchy-Riemann equations tell you then that $a=d$ and that $c=-b$. So, take $z_1=a+bi$ and $z_0=e+fi$. Then, $F(z)=z_1\times z+z_0$.

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