# Holomorphic Functions

• Kunihiko Kodaira
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 283)

## Abstract

We begin by defining holomorphic functions of n complex variables. The n-dimensional complex number space is the set of all n-tuples (z1,…, z n ) of complex numbers z i , i = 1,…, n, denoted by ℂ n . ℂ n is the Cartesian product of n copies of the complex plane: ℂ n = ℂ × … × ℂ. Denoting (z1,…, z n ) by z, we call z = (z1,…, z n ) a point of ℂ n , and zl,…, z n the complex coordinates of z. Letting z j = x2j−1 + ix2j by decomposing z j into its real and imaginary parts (where $$i = \sqrt { - 1}$$), we can express z as
$$z = ({x_1},{x_2}, \ldots ,{x_{2n - 1}},{x_{2n}}).$$
(1.1)

## Keywords

Power Series Holomorphic Function Analytic Continuation Power Series Expansion Irreducible Factorization
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.