Some Results and Problems on Complex Germs with Definable Mittag–Leffler Stars

Abstract

Working in an o-minimal expansion of the real field, we investigate when a germ (around zero, say) of a complex analytic function has a definable analytic continuation to its Mittag–Leffler star.

As an application we show that any algebro-logarithmic function that is complex analytic in a neighborhood of the origin in $\mathbb{C}$ has an analytic continuation to all but finitely many points in $\mathbb{C}$.