In the definition of an holomorphic function, it is said that one way of seeing this is by the existence of a variable $l$ (namely $f'(z_0)$) such that : $f(z) = f(z_0) + l(z-z_0) + o(|z-z_0|)$
If I recall correctly, this notation is equivalent to this one :
$\frac{f(z) - f(z_0) - l(z-z_0)}{|z-z_0|} \xrightarrow[z\to z_0]{} 0$
But that yields $sgn$ functions everywhere, doesn't it ?
I wonder if the module in the little-$o$ notation is neccessary ? And also what information does it give ?
Can one write : $f$ is holomorphic if one can find an $l\in\mathbb{C}$ such that $f(z) = f(z_0) + l(z-z_0) + o(z-z_0)$ ?
If I remember well, this kind of notation (with the absolute value) is present in real analysis too.