Definition of holomorphic functions in multiple dimensions

Question

What is the definition of a function $$f:U\rightarrow\mathbb{C}^n$$ being holomorphic? Where $U\subseteq\mathbb{C}^n$.

When I look around online all I can see is the definition for $$f:U\rightarrow\mathbb{C}$$ to be holomorphic. Such as $f$ being holomorphic in each variable $z_i$ (i.e. $\frac{\partial f}{\partial \bar{z_i}}=0$ where $\frac{\partial }{\partial \bar{z_i}}$ is the Wirtinger derivative.

Is it associated with the concept of total derivative?

Answer 1

$f : \Bbb{C^n \to C^m}$ is holomorphic on an open set $U$ iff for every $a \in U,v\in \Bbb{C^n}$, $D_{iv} f(a) = i D_v f(a)$ where $D_v f(a) = \lim_{t \to 0,t \in \Bbb{R}} \frac{f(a+tv)-f(a)}{t}$ is the directional derivative.

$U$ must be open in the complex topology.

The directional derivative is $\Bbb{R}$-linear in $v$ whenever $f$ is differentiable on $U$, holomorphy means it is $\Bbb{C}$-linear.

For $m=n$ the holomorphy means the Jacobian matrix of $(\Re(f),\Im(f))$ lies in the subalgebra of $M_{2n}(\Bbb{R})$ corresponding to $M_n(\Bbb{C})$.

For simplicity let $U \subset \Bbb{C}^2$, $f$ be holomorphic $U\to \Bbb{C}$ and $(a,b) \in U$ and $U$ be containing $\{ (u,v)\in \Bbb{C^2}, |u-a| \le r, |v-b| \le R\}$ then by the usual Cauchy integral formula

for $(z,w)\in \Bbb{C^2},|z-a| < r, |w-b| < R$ $$f(z,w) = \frac{1}{2i\pi}\int_{|u-a|=r} \frac{f(u,w)}{u-z}du=\frac{1}{(2i\pi)^2}\int_{|u-a|=r} \int_{|v-b|=R}\frac{f(u,v)}{(u-z)(v-w)}dudv$$ which is analytic by expanding the denominators in geometric series in $z,w$.