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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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I really wish you'd have mentioned the book you're looking at. My understanding from the limited information you gave is that an algebraic singularity is the type of singularity exhibited by a radical function like $\sqrt[n]{x-a}$ at $x=a$. – J. M. Mar 19 '12 at 11:47
@J.M. unfortunately it's not a book, but a not public text for us students (and it doesn't provide more details). Your guess makes sense and it bears me out that this is not a common term. I'll try to find my professor and ask him. I'll post the answer here. – Kiomse Mar 19 '12 at 11:52
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. – Zhen Lin Mar 19 '12 at 12:40
up vote 1 down vote accepted

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.

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