# Constructing the Koenigs function about a repelling fixed point

My question is rather simple and I hope someone has some sort of an answer. I am looking for a simple yes or no answer, and a reference if anyone has one.

We have a holomorphic function $f$ defined on some infinite subset of $\mathbb{C}$ and sends to this set. If $f(z_0) = z_0$ and $|f'(z_0)| > 1$ does the following limit construct the Koenigs function $\Psi$? Such that $\Psi(f(z)) = f'(z_0) \Psi(z)$ for $z$ in a sufficiently small enough neighborhood of $z_0$.

$$\lim_{n\to\infty} \frac{f^{\circ n}(z) - z_0}{f'(z_0)^n}$$

I'm well aware for the attracting case this is true (when 0 < |f'(z_0)| < 1), but I am unsure of the repelling case. I have seen it mentioned that this still holds, but I don't trust the source.

Perhaps someone has a proof this happens, or can link to a proof. Or simply give me a reason this doesn't happen. I'm at my wits end on how to prove or disprove this.

Thanks a whole bunch. I hope somebody can straighten this out for me.

Oh, if the absolute value of the derivative is not $1,$ no troubles.