The Wirtinger calculus , and its generalization by the Dolbeault operators , is a mathematical calculus from function theory . The Wirtinger calculus is named after the mathematician Wilhelm Wirtinger and the Dolbeault operators are named after Pierre Dolbeault . With the help of these objects, the representation of complex derivations can be made clearer. The Dolbeault operators are also used in the theory of quasi-conformal mappings .
Wirtinger calculation
A complex number is split into two real numbers by. Let be a domain and a (real) differentiable function. Then the partial derivatives exist
and
-
.
In the next section the Wirtinger derivatives are introduced, which are also partial differential operators. However, these are easier to calculate since the complex-valued function does not have to be broken down into real and imaginary parts. Instead of the coordinates and one uses and .
Motivation and Definition
With the help of the partial derivatives, the (total) differential of as is written
-
.
From and results
-
and .
For the differentials one obtains from this
-
and .
Inserting in the total differential and rearranging supplies
-
.
To (formally) the relationship
to get, you bet
and
-
.
These are the Wirtinger derivatives .
For you write short , for
you write . The operator is called the Cauchy-Riemann operator .
Holomorphic functions
The Wirtinger calculus is particularly used in function theory, since the notation for holomorphic functions is reduced to a minimum. In addition, this calculus is very stable, as properties 3 and 4 show in the next section.
A real differentiable function is a holomorphic function if and only if holds. In this case the derivative of . This is because the equation is a very brief representation of the Cauchy-Riemann differential equations . For this reason the operator is called the Cauchy-Riemann operator.
If, on the other hand, the equation applies to a real differentiable function, then this function is called antiholomorphic and the real differential can be calculated using property 1 .
properties
Relationship to partial derivative
The equations apply
and
-
.
Linearity
The operators and are - linear , that is for and real differentiable functions applies
and
-
.
Complex conjugation
For every real differentiable function holds
and
-
.
Chain rule
The chain rule applies to the Wirtinger derivatives
and
-
.
Main symbol
The main symbol of is and the main symbol of is . So both differential operators are elliptic .
Associated Laplace and Dirac operators
With the Wirtinger derivatives one can use the Laplace operator through
represent. In particular, it follows that the operator
is a Dirac operator .
Fundamental solution
The fundamental solution of the Cauchy-Riemann operator is , that is, the distribution generated by the function solves the equation , where is the delta distribution . A derivation can be found in the article Cauchy-Riemann partial differential equations .
Dolbeault operator
With the help of the Wirtinger calculus, one can also investigate multi-dimensional images. As above, elements of are decomposed into . Let now be an open subset and a (real) differentiable map. For this one defines the partial differential operators similar to the Wirtinger calculus
and
on . With the help of these partial differential operators one can find the Dolbeault operator and the Dolbeault transverse operator
and
define. These can be understood as multi-dimensional Wirtinger derivatives and are therefore noted in the same way. In addition, the Dolbeault operators have similar properties to the Wirtinger derivatives. In particular, it also holds that it is holomorphic if and only if and the real derivative becomes through
shown. In the holomorphic case it holds that yes .
Dolbeault operators on manifolds
The Dolbeault operator and the Dolbeault transverse operator can also be defined on complex manifolds , but the calculus of the complex differential forms must first be defined. With the help of the Dolbeault transverse operator, one can define holomorphic differential forms in the same way as in the previous section. One of the most important applications of these operators is to be found in the Hodge theory, especially in the Dolbeault cohomology , which is the complex analogue of the De Rham cohomology .
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