Wirtinger derivatives

In the Wirtinger calculus, and its generalization by the Dolbeault operators, it is a mathematical calculus of the theory of functions. The Wirtinger calculus is named after the mathematician Wilhelm Wirtinger and the Dolbeault operators are named after Pierre Dolbeault. Using these objects, the representation of complex derivatives can be made ​​clearer. In addition, the Dolbeault operators find application in the theory of quasi -conformal mappings.

Wirtinger calculus

Introduction

A complex number is divided by two real numbers. Be a territory and a ( real) differentiable function. Then there exist the partial derivatives

And

In the next section the Wirtinger - derivatives now be introduced, which are also partial differential operators. However, they are easier to calculate, since the complex-valued function need not be divided into real and imaginary parts. Instead of the coordinates and using and.

Motivation and definition

With the help of the partial derivatives to write the (total ) differential of as

From and results

For the differentials, one obtains

Provides insertion into the total differential and re-sorting

To ( formally) the relationship

To obtained, a

And

These are the Wirtinger - derivatives.

For we also write short, for you write. The operator is called the Cauchy -Riemann operator.

Holomorphic functions

The Wirtinger calculus finds particular application in the theory of functions, as the notation for holomorphic functions is reduced to a minimum. In addition, this calculation 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 the following holds. In this case, the discharge of. This is true because the equation is a very brief statement of the Cauchy- Riemann differential equations. For this reason, the operator is called Cauchy- Riemann operator.

Applies the other hand, for a real-differentiable function, the equation so you call this function antiholomorph and the real differential can be calculated with the help of property from one.

Other properties

  • Apply the equations
  • The operators and are - linearly, that is for real and differentiable functions applies
  • For each real-differentiable function
  • For the Wirtinger - derivatives, the chain rule
  • The main symbol of, and is the main symbol of. Both differential operators are therefore elliptical.
  • With the Wirtinger - derivatives can be the Laplace operator by
  • From the previous characteristic it follows that the operator
  • The fundamental solution of the Cauchy -Riemann operator, that is the distribution generated by the function solves the equation, where the delta function is. A derivation is to find partial differential equations in the Cauchy- Riemann article.

Dolbeault operator

Using the Wirtinger calculus can also investigate multi-dimensional pictures. As above elements of decomposed into. Now let be an open subset and a ( real) differentiable map. Therefore one is the the Wirtinger calculus similar partial differential operators

And

On. With the help of this partial differential operators can be the Dolbeault operator and the Dolbeault - cross - operator by

And

Define. This can be understood as multidimensional Wirtinger - derivatives and are therefore listed as well. In addition, the Dolbeault operators have similar characteristics as the Wirtinger - derivatives. In particular, also applies that is if and only holomorphic if and only if and the real derivation is

Shown. In the holomorphic case is valid, since valid.

Dolbeault operators on manifolds

The Dolbeault operator and the Dolbeault - cross - operator can be defined on complex manifolds, but it must first be defined the calculus of complex differential forms. Using the Dolbeault - cross - operator one can analogously define holomorphic differential forms as in the previous section. One of the most important applications of these operators can be found in Hodge theory, in particular in the Dolbeault cohomology, which is the complex analogue of the De Rham cohomology.

170435
de