\documentclass{amsart} \usepackage{amssymb} \usepackage[centertags]{amsmath} \usepackage{amssymb} \usepackage{xr} \usepackage{amscd} \usepackage{amsthm} \usepackage{amsxtra} \newcommand{\holo}[1]{\ensuremath{{ \mathcal O}({#1})} } \newcommand{\envelope}[1]{\ensuremath{\widetilde{#1}} } \newcommand{\re}{\ensuremath{\text{Re }}} \newcommand{\im}{\ensuremath{\text{Im }}} \theoremstyle{definition} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{cla}[theorem]{Claim} \newtheorem{problem}[theorem]{Problem} \newtheorem{definition}[theorem]{Definition} \begin{document} \title{$\mathbb{C}$ or $\mathbb{C}^n$? Envelopes of holomorphy.} \maketitle We discuss one of the major differences between complex analysis of one variable and complex analysis of several variables: the extension of holomorphic functions from domains to larger sets. We will provide motivation from the one variable theory, mention an open problem, and discuss the notions of domain of holomorphy'' and envelope of holomorphy''. We will also outline the construction of envelopes of holomorphy, illustrating three or four different approaches. \end{document}