operator

Let $G\subset {ℂ}^n$ be a domain and let$f:G\to ℂ$ be a $C^1$function (continuously differentiable)$(z^1,\mathrm{\dots },z^n)\mapsto f(z^1,\mathrm{\dots },z^n)$ where $z^j=x^j+i{y}^j$.We can think of $G$ as a subset of ${ℝ}^{2n}$.We thereforehave the following partial derivatives for all $1\le j\le n$,

${\displaystyle \frac{\partial f}{\partial z^j}}$$:={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial f}{\partial x^j}}-i{\displaystyle \frac{\partial f}{\partial y^j}}\right),$

${\displaystyle \frac{\partial f}{\partial {\overline{z}}^j}}$$:={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial f}{\partial x^j}}+i{\displaystyle \frac{\partial f}{\partial y^j}}\right).$

Now let $d$ be the standard exterior derivative on${ℝ}^{2n}$ and the $d{x}^j$ and $d{y}^j$ the standard basis of cotangentvectors. Then if we define

$d{z}^j$$:=d{x}^j+id{y}^j,$

$d{\overline{z}}^j$$:=d{x}^j-id{y}^j,$

then we can define two new operators acting on $C^1$ functions on $G$giving 1-forms by

$\partial f$$:={\displaystyle \sum _{j=1}^n}{\displaystyle \frac{\partial f}{\partial z^j}}d{z}^j,$

$\overline{\partial }f$$:={\displaystyle \sum _{j=1}^n}{\displaystyle \frac{\partial f}{\partial {\overline{z}}^j}}d{\overline{z}}^j.$

By direct calculation we immediately see that

$$df=\partial f+\overline{\partial }f.$$

Similarly we now define $\partial$ and $\overline{\partial }$on arbitrary differential form$\omega ={\sum }_{\alpha ,\beta }{f}_{\alpha ,\beta }d{z}^{\alpha }\wedge d{\overline{z}}^{\beta }$, where $\alpha$ and $\beta$ range over all multi-indices withelements less then $n$, where if $\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_k)$then $d{z}^{\alpha }=d{z}^{{\alpha }_1}\wedge \mathrm{\dots }\wedge d{z}^{{\alpha }_k}$,and $f_{\alpha ,\beta }$ is a $C^1$, complex valued functionon $G$.

$\partial \omega$$:={\displaystyle \sum _{\alpha ,\beta }}{\displaystyle \frac{\partial f_{\alpha ,\beta }}{\partial z^j}}d{z}^j\wedge d{z}^{\alpha }\wedge d{\overline{z}}^{\beta },$

$\overline{\partial }\omega$$:={\displaystyle \sum _{\alpha ,\beta }}{\displaystyle \frac{\partial f_{\alpha ,\beta }}{\partial {\overline{z}}^j}}d{\overline{z}}^j\wedge d{z}^{\alpha }\wedge d{\overline{z}}^{\beta }.$

Again a direct calculation shows that $d=\partial +\overline{\partial }$.

The Cauchy-Riemann equations are then given by

$$\overline{\partial }f=0$$

That is, $f$ is holomorphic if and only if it satisfies the above equations.Note that this only applies to functions. If $\overline{\partial }\omega =0$for a differential form, then the coefficients in the standard basisneed not be holomorphic.

Proposition.

$\overline{\partial }$ and $\mathrm{\partial }$ satisfy the following properties

• $\overline{\partial }$ and $\partial$ are linear,

• ${\overline{\partial }}^2=\overline{\partial }\overline{\partial }=0$ and ${\partial }^2=\partial \partial =0$,

• $\overline{\partial }\partial -\partial \overline{\partial }=0$.

While $\overline{\partial }u=0$ is our condition for $u$ to be aholomorphic function it turns out that it is more important to solve the inhomogeneous$\overline{\partial }u=f$ equation, as that allows us to construct holomorphicobjects from nonholomorphic ones.