Patrick Speissegger
This post describes the construction of quasianalytic algebras of functions with simple logarithmic transseries as asymptotic expansions. The construction is based on Ilyashenko’s class of almost regular functions, as introduced in his book on Dulac’s problem. This class forms a group under composition, but it is not closed under addition or multiplication; to obtain a…
(Joint work with Tobias Kaiser) Recall from this post that not all germs in $\H$ have a holomorphic extension that maps definable real domains to definable real domains. In fact, the extension $\t_a$ of the translation $t_a$, for $a\gt 0$, does not even map real domains to real domains. So, in order to describe the…
(Joint work with Tobias Kaiser) The goal of this post is to introduce a rough measure of size for a real domain $U$, based on the level of its boundary function $f_U$. As before, “definable” means “definable in $\Ranexp$”. Let $\I$ be the set of all infinitely increasing $\,f \in \H$ and set $\bo:= \H_{\gt…
(Joint work with Tobias Kaiser) I introduce here the types of domains in $\LL$ used later to describe holomorphic extensions of one-variable functions definable in $\Ranexp$. In this post “definable” means “definable in $\Ranexp$”. DefinitionA set $U \subseteq \LL$ is a real domain if there exist $a \gt 0$ and a continuous function $f =…
(Joint work with Tobias Kaiser) We are interested in holomorphic extensions of one-variable functions definable in $\Ranexp$. Since $\exp$ and $\log$ are two crucial functions definable in $\Ranexp$, the natural domain on which to consider holomorphic extensions of all definable functions is the Riemann surface of the logarithm $$\LL:= (0,\infty) \times \RR$$ with its usual…