We wish to study those domains in $\\mathbb{C}^n$, for $n\\geq 2$, the so-called domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We shall demonstrate that this study is radically different from that of domains in $\\mathbb{C}$ by discussing some examples of special types of domains in $\\mathbb{C}^n$, $n\\geq 2$, such that every function holomorphic on them extends to a strictly larger domain. This leads to Thullenâ€™s construction of a domain (not necessarily in $\\mathbb{C}^n$) spread over $\\mathbb{C}^n$, the so-called envelope of holomorphy, which fulfills our criteria. With the help of this abstract approach we shall give a characterization of the domains of holomorphy in $\\mathbb{C}^n$.

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Last updated: 06 Mar 2020