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Let $f : \Omega \to \mathbb{C}$ be holomorphic, where $\Omega$ is an annulus centered at $z=0$. We say that $f$ has a logarithmic singularity at $z=0$ if no analytic continuation of the germ of $f$ near $0$ with non-zero winding number about $0$ recovers $f$. Clearly, $\log z$ (for any branch) gives us an example of such a function.

Another example comes from $z^\alpha$, with $\alpha \in \mathbb{R} \smallsetminus \mathbb{Q}$. To contrast this, the branch point at $z=0$ is algebraic for $\alpha \in \mathbb{Q}$. In this case, we may express $f(z)$ locally as a Puiseux series (i.e. a power series in $1/p$, for some $p \in \mathbb{N}$). Do analogous series representations exist for logarithmic singularities, or are asymptotic expansions the best one can hope for?

Edit: As per Gerry Myerson's comment, take $\Omega$ as a sector with vertex removed.

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Not sure I understand. Neither $\log z$ nor $z^{\alpha}$ is holomorphic in an annulus centered at $z=0$. –  Gerry Myerson Feb 27 '12 at 23:42
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