Iss’sa’s Theorem. Domains of Holomorphy

Abstract

We begin by giving two interesting applications of the Weierstrass product theorem that have not yet made their way into the German textbook literature. In Section 1 we discuss Iss’sa’s theorem, discovered only in 1965; in Section 2 we show — once directly and once with the aid of the product theorem — that every domain in ℂ is a domain of holomorphy. In Section 3 we conclude by discussing simple examples of functions whose domains of holomorphy have the form $\left\{ {z \in {\Bbb C}:\left| {q(z)} \right| < R} \right\},q \in {\Bbb C}[z]$; Cassini domains, in particular, are of this form.