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I'm reading text about generating functions and how to reveal their asymptotic behaviour by means of analysing their singularities. In this context the term "algebraic singularity" or in German "algebraische Singularität" is used. An example is given: $p(x)=\frac{1-\sqrt{1-4x}}{2}$.

It's clear to me what a singularity is, but I can't find what they mean with algebaic in this context.

Edit: Is this term used as synonym for essential singularities, i.e. singularities not being poles?

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    The example you mentioned is not an essential singularity, and not a pole. If I'm not mistaken the only remaining possibility is that it is a branch point. Algebraicity probably refers to the fact that the ramification index is finite. – 2012-03-19

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I think I'm now able to answer my question. Thanks to J.M. and Zhen Lin for their comments!

First of all let's clarify the situation of a square root at zero. I had in mind that there are exactly three types of points: holomorphic, poles and essential singularities. Well that's only half of the truth. This is valid if $x$ is a inner point of $G$ and $f$ holomorphic on $G \setminus \{x\}$. In this case $x$ is one of the three types mentioned before.

The root is not holomorphic defineable on any set $G \setminus \{0\}$ with $0$ being a inner point of $G$ and therefore it's a branching point and not a essential singularity.

That's why I thought every branching point must be a essential singularity (clearly its not holomorphic or a pole).

A algebraic singularity is defined as singularity of the type $z^{p/q}$ at $0$ for some integers p and q. The special case p=1 and q=2 is called square root singularity.