Complex Geometry and Kahler Geometry_20130227p1

0.1. Holomorphic Map

Definition 1. A complex-valued function $f(z)$ defined on a connected open domain $W\subseteq\C^n$ is called holomorphic if for each $a=(a_1,\ldots, a_n)\in W$, $f(z)$ can be represented as a power series
\[
\sum_{k_1\geq0,\cdots,k_n\geq0}^\infty C_{k_1\cdots k_n}(z_1-a_1)^{k_1}\cdots (z_n-a_n)^k_n
\]
in some neighbourhood of $a$.

We have the following equivalent definition Continue Reading