Let $n \in \mathbb{N}$, let $U \subset \mathbb{C}^n$ be open, and let $f: U \to \mathbb{C}$ be a function in $C^\infty(U;\mathbb{C})$, where smoothness is understood after identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$. For each $j \in \{1,\dots,n\}$, let $x_j:U\to\mathbb{R}$ and $y_j:U\to\mathbb{R}$ be the real coordinate maps determined by $z_j=x_j+i y_j$. Define the complex-valued smooth function $\partial f/\partial \bar z_j:U\to\mathbb{C}$ by

\begin{align*} \frac{\partial f}{\partial \bar z_j} := \frac{1}{2}\left(\frac{\partial f}{\partial x_j} + i\frac{\partial f}{\partial y_j}\right). \end{align*}

Let $d\bar z_1,\dots,d\bar z_n$ denote the standard anti-holomorphic coordinate one-forms on $U$. Define the $(0,1)$-form $\bar\partial f$ on $U$ by

\begin{align*} \bar\partial f := \sum_{j=1}^n \frac{\partial f}{\partial \bar z_j}\, d\bar z_j. \end{align*}

The equation $\bar\partial f=0$ on $U$ means that $\partial f/\partial \bar z_j(a)=0$ for every $a\in U$ and every $j\in\{1,\dots,n\}$. Here $f$ is holomorphic on $U$ means that for every $a \in U$ there exists a complex-[linear map](/page/Linear%20Map) $L_a: \mathbb{C}^n \to \mathbb{C}$ such that

\begin{align*} f(a+h)=f(a)+L_a(h)+o(|h|) \quad \text{as } h \to 0 \text{ in } \mathbb{C}^n \text{ with } a+h\in U. \end{align*}

Then $f$ is holomorphic on $U$ if and only if $\bar\partial f = 0$ on $U$.