Editing Holomorphic function (section)

== Several variables ==
The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function {{tmath|f \colon ( z_1, z_2, \ldots, z_n ) \mapsto f( z_1, z_2, \ldots, z_n ) }} in {{tmath|n}} complex variables is '''analytic''' at a point {{tmath|p}} if there exists a neighbourhood of {{tmath|p}} in which {{tmath|f}} is equal to a convergent power series in {{tmath|n}} complex variables;<ref>
{{cite book
 |last1=Gunning |last2=Rossi |name-list-style=and
 |title=Analytic Functions of Several Complex Variables
 |page=2
}}
</ref>
the function {{tmath|f}} is '''holomorphic''' in an open subset {{tmath|U}} of {{tmath|\C^n}} if it is analytic at each point in {{tmath|U}}.  [[Osgood's lemma]] shows (using the multivariate Cauchy integral formula) that, for a continuous function {{tmath|f}}, this is equivalent to {{tmath|f}} being holomorphic in each variable separately (meaning that if any {{tmath|n-1}} coordinates are fixed, then the restriction of {{tmath|f}} is a holomorphic function of the remaining coordinate).  The much deeper [[Hartogs' theorem]] proves that the continuity assumption is unnecessary: {{tmath|f}} is holomorphic if and only if it is holomorphic in each variable separately.

More generally, a function of several complex variables that is [[square integrable]] over every [[compact set|compact subset]] of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.

Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex [[Reinhardt domain]]s, the simplest example of which is a [[polydisk]].  However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called a [[domain of holomorphy]].

A [[complex differential form#Holomorphic forms|complex differential {{tmath|(p,0)}}-form]] {{tmath|\alpha}} is holomorphic if and only if its antiholomorphic [[Complex differential form#The Dolbeault operators|Dolbeault derivative]] is zero: {{tmath|1= \bar{\partial}\alpha = 0}}.