Editing Analytic function (section)

==Analytic functions of several variables==
One can define analytic functions in several variables by means of power series in those variables (see [[power series]]). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in 2 or more complex dimensions:

* Zero sets of complex analytic functions in more than one variable are never [[discrete space|discrete]]. This can be proved by [[Hartogs's extension theorem]].
* [[Domain of holomorphy|Domains of holomorphy]] for single-valued functions consist of arbitrary (connected) open sets. In several complex variables, however, only some connected open sets are domains of holomorphy. The characterization of domains of holomorphy leads to the notion of [[pseudoconvexity]].