holomorphic functions of several variables

Definition.

Let $\mathrm{\Omega }\subset {ℂ}^n$ be a domain and let $f:\mathrm{\Omega }\to ℂ$ be a function. $f$ is called if it isholomorphic (http://planetmath.org/Holomorphic) in each variable separately as a function of one variable.

That means that the function $z_k\mapsto f(z_1,\mathrm{\dots },z_k,\mathrm{\dots },z_n)$is holomorphic as a function of one variable. It is not at all obvious thatsuch a function is even continuous and we must apply the Hartogs’s theorem on separateanalyticity which is not a trivial result.

Historically and some authors today still continue to do so, the definitionof being holomorphic in several variables did include the continuity or atleast local boundedness requirement.

Of course we can also characterize holomorphic functions by their power series.

Proposition.

$f$ is holomorphic in $\mathrm{\Omega }$ if and only if near each point $\zeta \mathrm{\in }\mathrm{\Omega }$ there is a neighbourhood$U$ and a power series in several variables

$$\sum _{\alpha }{a}_{\alpha }{(z-\zeta )}^{\alpha },$$

where $\alpha$ ranges over all themulti-indices, $a_{\alpha }\mathrm{\in }\mathrm{C}$ and such that the series converges to $f\mathit{}\mathrm{(}z\mathrm{)}$ for $z\mathrm{\in }U$.

Another way to characterize holomorphic functions is by the use of theCauchy-Riemann equations, which can be given in a very form by the$\overline{\partial }$-operator (http://planetmath.org/BarpartialOperator).

Proposition.

$f$ is holomorphic if and only if $\overline{\mathrm{\partial }}\mathit{}f\mathrm{=}\mathrm{0}$.

Despite the similarities,one should be careful about carelessly generalizing results about functionsof one variable to functions of several variables as the theory is quitedifferent. See the topic entry on several complex variables for more.