On the Existence of Holomorphic Functions Having Prescribed Asymptotic Expansions

Abstract

A generalization of some results of T. Carleman in [1] is developped. The practical form of it states that if the non-empty subset $D$ of the boundary $\partial \Omega$ of a domain $\Omega$ of $C$ has no accumulation point and if the connected component in $\partial \Omega$ of every is $u\in D$ has more than one point, then $D$ is regularly asymptotic for $\Omega$, i.e. for every family {$c_{u,n}:u\in D,n\in {\mathbb{N}}_0$} of complex numbers, there is a holomorphic function $f$ on $\Omega$ which at every $u\in D$ has ${\sum }_{n=0}^{\infty }{c}_{u,n}(z–u{)}^n$ as asymptotic expansion at $u$.