Abstract
Arakelian’s classical approximation theorem (N. U. Arakelian, Izv. Akad. Nauk SSSR 28, 1187–1206 (1964)) gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets \(F\subset\mathbb{C}\) by entire functions. The conditions are purely topological and concern the connectedness of the complement of \(F\) . We give a new characterization of Arakelian sets in terms of logarithmic branches of functions \(f\in A(F)\) , which are continuous in \(F\) and holomorphic in its interior \(F^{\circ}\) . Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem.
