My book (Real and complex analysis, by Rudin) gives the following definition:
Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + re^{i\theta} )- f(z_0)] \ \ \ \ (r > 0)$$ exists and it is independent of $\theta$.
It then adds
The requirements is that fr any two rays $L'$ and $L''$ starting at $z_0$, the angles which their images $f(L')$ and $f(L'')$ make at $f(z_0)$ is the same as that made by $L'$ and $L''$ in size as well orientation
I am trying to understand why the definition says that. Does anyone know of a cool geometric interpretation?
My thoughts
$A(z)$ returns a complex number on the unit circle whose argument is the argument of $z$. So $A(z - w)$ basically is a measure of the angle between $z$ and $w$.
Now in the definition I take the angle between $f(z_0 + re^{i \theta})$ and $f(z_0)$, and divide it by $e^{i \theta}$ (meaning I rotate by $-\theta$ our number). Since the original angle between $z_0$ and $z_0 + re^{i \theta}$ is $\theta$, if I want angles to be preserved, I would like the result to have an argument of $0$, right? Why is instead required the limit to exists and be independent of $\theta$?
Moreover, the reason why we take the limit is because we only care about local properties of the function, right?
Finally, why was it defined this way? I mean one could be naive and define it like $$\lim_{r \to 0} \arg(f(z_0 + re^{i\theta}) - f(z_0)) = \theta$$. What's the reason this isn't a good definition for our purposes?
Your thougts are almost spot on, save for a little detail: $A(z-w)$ does not measure the argument between $z$ and $w$, it actually measures the argument of $z$ after being translated by $w$ (which is not the same thing as there is no rotation involved in translating something by $w$, I recommend you make a drawing to convince yourself of this fact).
So in fact $A(f(z + r e^{i\theta}) - f(z))$ measures the argument of $f(z+re^{i\theta})$ when we regard $f(z)$ as the origin.
And I guess your alternative definition is correct, but the one in the book suggests a more concrete analytical treatment.