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 operator
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(Definition)
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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
$1 \leq j \leq n$ ,
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
then we can define two new operators acting on $C^1$ functions on $G$ giving 1-forms by
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 $G$ .
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.
Proposition 1 $\bar{\partial}$ and $\partial$ satisfy the following properties
- $\bar{\partial}$ and $\partial$ are linear,
- $\bar{\partial}^2 = \bar{\partial} \bar{\partial} = 0$ and $\partial^2 = \partial \partial = 0$ ,
- $\bar{\partial} \partial - \partial \bar{\partial} = 0$ .
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.
- 1
- Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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" operator" is owned by jirka.
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(view preamble | get metadata)
| Other names: |
d bar operator, d-bar operator |
| Also defines: |
 operator |
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Cross-references: objects, inhomogeneous, properties, coefficients, equations, holomorphic, Cauchy-Riemann equations, complex, multi-indices, range, differential form, 1-forms, operators, vectors, cotangent, standard basis, exterior derivative, partial derivatives, subset, continuously differentiable, function, domain
There are 4 references to this entry.
This is version 4 of  operator, born on 2005-04-05, modified 2005-11-03.
Object id is 6931, canonical name is BarpartialOperator.
Accessed 5046 times total.
Classification:
| AMS MSC: | 30E99 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Miscellaneous) | | | 32A99 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Miscellaneous) |
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Pending Errata and Addenda
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