BarpartialOperator 2005-04-05 20:22:45 2005-11-03 19:43:02 Definition $\partial$ operator % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{theorem} \newtheorem*{thm}{Theorem} \newtheorem*{lemma}{Lemma} \newtheorem*{conj}{Conjecture} \newtheorem*{cor}{Corollary} \newtheorem*{example}{Example} \newtheorem*{prop}{Proposition} \theoremstyle{definition} \newtheorem*{defn}{Definition} \theoremstyle{remark} \newtheorem*{rmk}{Remark} Let $G \subset {\mathbb{C}}^n$ be a domain and let $f \colon G \to {\mathbb{C}}$ be a $C^1$ function (continuously differentiable) $(z^1,\ldots,z^n) \mapsto f(z^1,\ldots,z^n)$ where $z^j = x^j + i y^j$. We can think of $G$ as a subset of ${\mathbb{R}}^{2n}$. We therefore have the following partial derivatives for all $1 \leq j \leq n$, \begin{align*} \frac{\partial f}{\partial z^j} & := \frac{1}{2} \left( \frac{\partial f}{\partial x^j} - i \frac{\partial f}{\partial y^j} \right) , \\ \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*} Now let $d$ be the standard exterior derivative on ${\mathbb{R}}^{2n}$ and the $dx^j$ and $dy^j$ the standard basis of cotangent vectors. Then if we define \begin{align*} dz^j & := dx^j + i dy^j , \\ d\bar{z}^j & := dx^j - i dy^j , \end{align*} then we can define two new operators acting on $C^1$ functions on $G$ giving 1-forms by \begin{align*} \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} d\bar{z}^j . \end{align*} By direct calculation we immediately see that \begin{equation*} df = \partial f + \bar{\partial} f . \end{equation*} Similarly we now define $\partial$ and $\bar{\partial}$ on arbitrary differential form $\omega = \sum_{\alpha,\beta} f_{\alpha,\beta} dz^\alpha \wedge d\bar{z}^\beta$, where $\alpha$ and $\beta$ range over all multi-indices with elements less then $n$, where if $\alpha = (\alpha_1,\ldots,\alpha_k)$ then $dz^\alpha = dz^{\alpha_1} \wedge \ldots \wedge dz^{\alpha_k}$, and $f_{\alpha,\beta}$ is a $C^1$, complex valued function on $G$. \begin{align*} \partial \omega & := \sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial z^j} dz^j \wedge dz^\alpha \wedge d\bar{z}^\beta , \\ \bar{\partial} \omega & := \sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial \bar{z}^j} d\bar{z}^j \wedge dz^\alpha \wedge d\bar{z}^\beta . \end{align*} Again a direct calculation shows that $d = \partial + \bar{\partial}$. The Cauchy-Riemann equations are then given by \begin{equation*} \bar{\partial} f = 0 \end{equation*} That is, $f$ is holomorphic if and only if it satisfies the above equations. Note that this only applies to functions. If $\bar{\partial}\omega = 0$ for a differential form, then the coefficients in the standard basis need not be holomorphic. \begin{prop} $\bar{\partial}$ and $\partial$ satisfy the following properties \begin{itemize} \item $\bar{\partial}$ and $\partial$ are linear, \item $\bar{\partial}^2 = \bar{\partial} \bar{\partial} = 0$ and $\partial^2 = \partial \partial = 0$, \item $\bar{\partial} \partial - \partial \bar{\partial} = 0$. \end{itemize} \end{prop} 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 $\bar{\partial}u = f$ equation, as that allows us to construct holomorphic objects from nonholomorphic ones. \begin{thebibliography}{9} \bibitem{Hormander:several} Lars H\"ormander. {\em \PMlinkescapetext{An Introduction to Complex Analysis in Several Variables}}, North-Holland Publishing Company, New York, New York, 1973. \bibitem{Krantz:several} Steven~G.\@ Krantz. {\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, AMS Chelsea Publishing, Providence, Rhode Island, 1992. \end{thebibliography}