| Version current |
Version 3 |
| Let $G \subset {\mathbb{C}}^n$ be a domain and let |
Let $G \subset {\mathbb{C}}^n$ be a domain and let |
| $f \colon G \to {\mathbb{C}}$ be a $C^1$ |
$f \colon G \to {\mathbb{C}}$ be a $C^1$ |
| function (continuously differentiable) |
function (continuously differentiable) |
| $(z^1,\ldots,z^n) \mapsto f(z^1,\ldots,z^n)$ where $z^j = x^j + i y^j$. |
$(z^1,\ldots,z^n) \mapsto f(z^1,\ldots,z^n)$ where $z^j = x^j + i y^j$. |
| We can think of $G$ as a subset of ${\mathbb{R}}^{2n}$. |
We can think of $G$ as a subset of ${\mathbb{R}}^{2n}$. |
| We therefore |
We therefore |
| have the following partial derivatives for all $1 \leq j \leq n$, |
have the following partial derivatives for all $1 \leq j \leq n$, |
| \begin{align*} |
\begin{align*} |
| \frac{\partial f}{\partial z^j} & := |
\frac{\partial f}{\partial z^j} & := |
| \frac{1}{2} \left( |
\frac{1}{2} \left( |
| \frac{\partial f}{\partial x^j} - i \frac{\partial f}{\partial y^j} |
\frac{\partial f}{\partial x^j} - i \frac{\partial f}{\partial y^j} |
| \right) , |
\right) , |
| \\ |
\\ |
| \frac{\partial f}{\partial \bar{z}^j} & := |
\frac{\partial f}{\partial \bar{z}^j} & := |
| \frac{1}{2} \left( |
\frac{1}{2} \left( |
| \frac{\partial f}{\partial x^j} + i \frac{\partial f}{\partial y^j} |
\frac{\partial f}{\partial x^j} + i \frac{\partial f}{\partial y^j} |
| \right) . |
\right) . |
| \end{align*} |
\end{align*} |
| Now let $d$ be the standard exterior derivative on |
Now let $d$ be the standard exterior derivative on |
| ${\mathbb{R}}^{2n}$ and the $dx^j$ and $dy^j$ the standard basis of cotangent |
${\mathbb{R}}^{2n}$ and the $dx^j$ and $dy^j$ the standard basis of cotangent |
| vectors. Then if we define |
vectors. Then if we define |
| \begin{align*} |
\begin{align*} |
| dz^j & := dx^j + i dy^j , \\ |
dz^j & := dx^j + i dy^j , \\ |
| d\bar{z}^j & := dx^j - i dy^j , |
d\bar{z}^j & := dx^j - i dy^j , |
| \end{align*} |
\end{align*} |
| then we can define two new operators acting on $C^1$ functions on $G$ |
then we can define two new operators acting on $C^1$ functions on $G$ |
| giving 1-forms by |
giving 1-forms by |
| \begin{align*} |
\begin{align*} |
| \partial f & := \sum_{j=1}^n \frac{\partial f}{\partial z^j} dz^j , \\ |
\partial f & := \sum_{j=1}^n \frac{\partial f}{\partial z^j} dz^j , \\ |
| \bar{\partial} f & := \sum_{j=1}^n \frac{\partial f}{\partial \bar{z}^j} |
\bar{\partial} f & := \sum_{j=1}^n \frac{\partial f}{\partial \bar{z}^j} |
| d\bar{z}^j . |
d\bar{z}^j . |
| \end{align*} |
\end{align*} |
| By direct calculation we immediately see that |
By direct calculation we immediately see that |
| \begin{equation*} |
\begin{equation*} |
| df = \partial f + \bar{\partial} f . |
df = \partial f + \bar{\partial} f . |
| \end{equation*} |
\end{equation*} |
|
|
| Similarly we now define $\partial$ and $\bar{\partial}$ |
Similarly we now define $\partial$ and $\bar{\partial}$ |
| on arbitrary differential form |
on arbitrary differential form |
| $\omega = \sum_{\alpha,\beta} f_{\alpha,\beta} dz^\alpha \wedge |
$\omega = \sum_{\alpha,\beta} f_{\alpha,\beta} dz^\alpha \wedge |
| d\bar{z}^\beta$, where $\alpha$ and $\beta$ range over all multi-indices with |
d\bar{z}^\beta$, where $\alpha$ and $\beta$ range over all multi-indices with |
| elements less then $n$, where if $\alpha = (\alpha_1,\ldots,\alpha_k)$ |
elements less then $n$, where if $\alpha = (\alpha_1,\ldots,\alpha_k)$ |
|
then $dz^\alpha = dz^{\alpha_1} \wedge \ldots \wedge dz^{\alpha_k}$,
|
then $dz^\alpha = dz_{\alpha_1} \wedge \ldots \wedge dz^{\alpha_k}$,
|
| and $f_{\alpha,\beta}$ is a $C^1$, complex valued function |
and $f_{\alpha,\beta}$ is a $C^1$, complex valued function |
| on $G$. |
on $G$. |
| \begin{align*} |
\begin{align*} |
| \partial \omega |
\partial \omega |
| & := |
& := |
| \sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial z^j} dz^j |
\sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial z^j} dz^j |
| \wedge |
\wedge |
| dz^\alpha \wedge d\bar{z}^\beta |
dz^\alpha \wedge d\bar{z}^\beta |
| , \\ |
, \\ |
| \bar{\partial} \omega |
\bar{\partial} \omega |
| & := |
& := |
| \sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial \bar{z}^j} |
\sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial \bar{z}^j} |
| d\bar{z}^j |
d\bar{z}^j |
| \wedge |
\wedge |
| dz^\alpha \wedge d\bar{z}^\beta . |
dz^\alpha \wedge d\bar{z}^\beta . |
| \end{align*} |
\end{align*} |
| Again a direct calculation shows that $d = \partial + \bar{\partial}$. |
Again a direct calculation shows that $d = \partial + \bar{\partial}$. |
|
|
| The Cauchy-Riemann equations are then given by |
The Cauchy-Riemann equations are then given by |
| \begin{equation*} |
\begin{equation*} |
| \bar{\partial} f = 0 |
\bar{\partial} f = 0 |
| \end{equation*} |
\end{equation*} |
| That is, $f$ is holomorphic if and only if it satisfies the above equations. |
That is, $f$ is holomorphic if and only if it satisfies the above equations. |
| Note that this only applies to functions. If $\bar{\partial}\omega = 0$ |
Note that this only applies to functions. If $\bar{\partial}\omega = 0$ |
| for a differential form, then the coefficients in the standard basis |
for a differential form, then the coefficients in the standard basis |
| need not be holomorphic. |
need not be holomorphic. |
|
|
| \begin{prop} |
\begin{prop} |
| $\bar{\partial}$ and $\partial$ satisfy the following properties |
$\bar{\partial}$ and $\partial$ satisfy the following properties |
| \begin{itemize} |
\begin{itemize} |
| \item $\bar{\partial}$ and $\partial$ are linear, |
\item $\bar{\partial}$ and $\partial$ are linear, |
| \item $\bar{\partial}^2 = \bar{\partial} \bar{\partial} = 0$ and $\partial^2 = \partial \partial = 0$, |
\item $\bar{\partial}^2 = \bar{\partial} \bar{\partial} = 0$ and $\partial^2 = \partial \partial = 0$, |
| \item $\bar{\partial} \partial - \partial \bar{\partial} = 0$. |
\item $\bar{\partial} \partial - \partial \bar{\partial} = 0$. |
| \end{itemize} |
\end{itemize} |
| \end{prop} |
\end{prop} |
|
|
| While $\bar{\partial} u = 0$ is our condition for $u$ to be a |
While $\bar{\partial} u = 0$ is our condition for $u$ to be a |
| holomorphic function it turns out that it is more important to solve the inhomogeneous |
holomorphic function it turns out that it is more important to solve the inhomogeneous |
| $\bar{\partial}u = f$ equation, as that allows us to construct holomorphic |
$\bar{\partial}u = f$ equation, as that allows us to construct holomorphic |
| objects from nonholomorphic ones. |
objects from nonholomorphic ones. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Hormander:several} |
\bibitem{Hormander:several} |
| Lars H\"ormander. |
Lars H\"ormander. |
| {\em \PMlinkescapetext{An Introduction to Complex Analysis in Several |
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several |
| Variables}}, |
Variables}}, |
| North-Holland Publishing Company, New York, New York, 1973. |
North-Holland Publishing Company, New York, New York, 1973. |
| \bibitem{Krantz:several} |
\bibitem{Krantz:several} |
| Steven~G.\@ Krantz. |
Steven~G.\@ Krantz. |
| {\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, |
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, |
| AMS Chelsea Publishing, Providence, Rhode Island, 1992. |
AMS Chelsea Publishing, Providence, Rhode Island, 1992. |
| \end{thebibliography} |
\end{thebibliography} |
