Let $f: \mathbb{C} \to \hat{\mathbb{C}} := \mathbb{C} \cup \{ \infty \}$ a multi-valued function in the context of complex analysis with $f(0) = \infty$.
The question is why this function $f$ can be near $0$ locally expanded as
$$ g(z) + A \cdot log \ z +B z^{-1}+ C z^{-2}+ ... $$
with $g(z)$ finite and holomorphic near $0$, the $A, B, C,...$ appropriate complex comstanats and $log \ z$ the (multivalued) complex logarithm? So the question is what kind of branch point branch point a singularity of $f$ might have? The expansion about suggests that the only kind of branches may occure are the logarithmic ones if we assume $f(0) = \infty$ (ie that $0$ is singular point). That means that $0$ cannot be an algebraic or non logarithmic transcendental branch point. Why?
(here I implicitely assume that a multivalued function $h(z)$ with singuarity in $0$ logarithmically branches in $0$ iff locally $h= log \ z + g(z)$ where $g(z)$ is single-valued meromorphic function; is this definition of logarithmic branch correct? Looks reasonable to me, but haven't seen this definition in literature, so beside the concern of 'main question' I would like to know if that's what I wrote in previous sentence is a conventional definition of logarithmical branch)
The background of this question is an unproved statement in Riemann's 'Theorie der Abelschen Functionen' I intended to discuss in MathOverflow but there was attested to me too little basic knowledge. But I don't know any textbook on complex analysis which provide exactly that backgrond to discuss this broblem or where this problem is discussed. Nevertheless I humbly think that I already have a certain amount of basic knowledge on complex analysis. So possibly mse might be a better place to discuss this problem.