Holomorphic Functions

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 ℂⁿ. ℂⁿ is the Cartesian product of n copies of the complex plane: ℂⁿ = ℂ × … × ℂ. Denoting (z₁,…, z _n ) by z, we call z = (z₁,…, z _n ) a point of ℂⁿ, and z_l,…, z _n the complex coordinates of z. Letting z _j = x_2j−1 + ix_2j 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)