Wirtinger calculation

Wilhelm Wirtinger

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 ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle z:=x+\mathrm {i} y}$ ${\displaystyle G\subset \mathbb {C} }$ ${\displaystyle f=u+\mathrm {i} v\colon G\to \mathbb {C} }$

${\displaystyle {\frac {\partial f}{\partial x}}={\frac {\partial u}{\partial x}}+\mathrm {i} {\frac {\partial v}{\partial x}}}$

and

${\displaystyle {\frac {\partial f}{\partial y}}={\frac {\partial u}{\partial y}}+\mathrm {i} {\frac {\partial v}{\partial y}}}$.

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 . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z=x+\mathrm {i} y}$ ${\displaystyle {\bar {z}}=x-\mathrm {i} y}$

Motivation and Definition

With the help of the partial derivatives, the (total) differential of as is written ${\displaystyle f}$

${\displaystyle \mathrm {d} f={\frac {\partial f}{\partial x}}\mathrm {d} x+{\frac {\partial f}{\partial y}}\mathrm {d} y}$.

From and results ${\displaystyle z=x+\mathrm {i} y}$ ${\displaystyle {\bar {z}}=x-\mathrm {i} y}$

${\displaystyle \textstyle x={\frac {1}{2}}(z+{\bar {z}})}$and . ${\displaystyle \textstyle y={\frac {1}{2\mathrm {i} }}(z-{\bar {z}})={\frac {\mathrm {i} }{2}}({\bar {z}}-z)}$

For the differentials one obtains from this

${\displaystyle \mathrm {d} x={\frac {1}{2}}(\mathrm {d} z+\mathrm {d} {\bar {z}})}$and . ${\displaystyle \mathrm {d} y={\frac {\mathrm {i} }{2}}(\mathrm {d} {\bar {z}}-\mathrm {d} z)}$

Inserting in the total differential and rearranging supplies

${\displaystyle \mathrm {d} f={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-\mathrm {i} {\frac {\partial f}{\partial y}}\right)\mathrm {d} z+{\frac {1}{2}}\left({\frac {\partial f}{\partial x}}+\mathrm {i} {\frac {\partial f}{\partial y}}\right)\mathrm {d} {\bar {z}}}$.

To (formally) the relationship

${\displaystyle \mathrm {d} f={\frac {\partial f}{\partial z}}\mathrm {d} z+{\frac {\partial f}{\partial {\bar {z}}}}\mathrm {d} {\bar {z}}}$

to get, you bet

${\displaystyle {\frac {\partial f}{\partial z}}:={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-\mathrm {i} {\frac {\partial f}{\partial y}}\right)}$

and

${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}:={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}+\mathrm {i} {\frac {\partial f}{\partial y}}\right)}$.

These are the Wirtinger derivatives .

For you write short , for you write . The operator is called the Cauchy-Riemann operator . ${\displaystyle \textstyle {\frac {\partial f}{\partial z}}}$ ${\displaystyle \,\partial f}$ ${\displaystyle \textstyle {\frac {\partial f}{\partial {\bar {z}}}}}$ ${\displaystyle {\bar {\partial }}f}$ ${\displaystyle {\overline {\partial }}}$

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. ${\displaystyle {\overline {\partial }}f=0}$ ${\displaystyle \partial f}$ ${\displaystyle f}$ ${\displaystyle {\overline {\partial }}f=0}$ ${\displaystyle {\overline {\partial }}}$

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 . ${\displaystyle f}$ ${\displaystyle \partial f=0}$ ${\displaystyle {\overline {\partial }}f}$

properties

Relationship to partial derivative

The equations apply

${\displaystyle {\frac {\partial f}{\partial x}}=\partial f+{\overline {\partial }}f}$

and

${\displaystyle {\frac {\partial f}{\partial y}}=\mathrm {i} \left(\partial f-{\overline {\partial }}f\right)}$.

Linearity

The operators and are - linear , that is for and real differentiable functions applies ${\displaystyle \partial }$ ${\displaystyle {\overline {\partial }}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a,b\in \mathbb {C} }$ ${\displaystyle f,g\colon G\to \mathbb {C} }$

${\displaystyle \partial (af+bg)=a\partial f+b\partial g}$

and

${\displaystyle {\overline {\partial }}(af+bg)=a{\overline {\partial }}f+b{\overline {\partial }}g}$.

Complex conjugation

For every real differentiable function holds ${\displaystyle f}$

${\displaystyle {\overline {\partial }}f={\overline {\partial {\overline {f}}}}}$

and

${\displaystyle {\overline {\partial }}\ {\overline {f}}={\overline {\partial f}}}$.

Chain rule

The chain rule applies to the Wirtinger derivatives

${\displaystyle {\frac {\partial (g\circ f)}{\partial z}}(z_{0})={\frac {\partial g}{\partial w}}(f(z_{0}))\cdot {\frac {\partial f}{\partial z}}(z_{0})+{\frac {\partial g}{\partial {\overline {w}}}}(f(z_{0}))\cdot {\frac {\partial {\overline {f}}}{\partial z}}(z_{0})}$

and

${\displaystyle {\frac {\partial (g\circ f)}{\partial {\overline {z}}}}(z_{0})={\frac {\partial g}{\partial w}}(f(z_{0}))\cdot {\frac {\partial f}{\partial {\overline {z}}}}(z_{0})+{\frac {\partial g}{\partial {\overline {w}}}}(f(z_{0}))\cdot {\frac {\partial {\overline {f}}}{\partial {\overline {z}}}}(z_{0})}$.

Main symbol

The main symbol of is and the main symbol of is . So both differential operators are elliptic . ${\displaystyle \partial }$ ${\displaystyle \xi \mapsto {\tfrac {1}{2}}(\xi _{1}-\mathrm {i} \xi _{2})}$ ${\displaystyle {\overline {\partial }}}$ ${\displaystyle \xi \mapsto {\tfrac {1}{2}}(\xi _{1}+\mathrm {i} \xi _{2})}$

Associated Laplace and Dirac operators

With the Wirtinger derivatives one can use the Laplace operator through

${\displaystyle \Delta f=4\partial {\overline {\partial }}f=4{\overline {\partial }}\partial f}$

represent. In particular, it follows that the operator

${\displaystyle D:=2{\begin{pmatrix}0&-\partial \\{\overline {\partial }}&0\end{pmatrix}}}$

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 . ${\displaystyle \textstyle {\frac {\partial }{\partial {\overline {z}}}}}$ ${\displaystyle \textstyle {\frac {1}{\pi z}}}$ ${\displaystyle \textstyle u(z)={\frac {1}{\pi z}}}$ ${\displaystyle \textstyle {\frac {\partial }{\partial {\overline {z}}}}u(z)=\delta }$ ${\displaystyle \delta }$

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 ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle (z_{1},\ldots z_{n})=(x_{1}+\mathrm {i} y_{1},\ldots ,x_{n}+\mathrm {i} y_{n})}$ ${\displaystyle D\subset \mathbb {C} ^{n}}$ ${\displaystyle f=(f_{1},\ldots ,f_{m}):D\rightarrow \mathbb {C} ^{m}}$

${\displaystyle {\frac {\partial }{\partial z_{j}}}:={\frac {1}{2}}\left({\frac {\partial }{\partial x_{j}}}-\mathrm {i} {\frac {\partial }{\partial y_{j}}}\right)\quad j=1,\ldots ,n}$

and

${\displaystyle {\frac {\partial }{\partial {\bar {z}}_{j}}}:={\frac {1}{2}}\left({\frac {\partial }{\partial x_{j}}}+\mathrm {i} {\frac {\partial }{\partial y_{j}}}\right)\quad j=1,\ldots ,n}$

on . With the help of these partial differential operators one can find the Dolbeault operator and the Dolbeault transverse operator ${\displaystyle \mathbb {C} ^{n}}$

${\displaystyle \partial f:=\sum _{j=1}^{n}{\frac {\partial }{\partial z_{j}}}f{\rm {d}}z_{j}}$

and

${\displaystyle {\overline {\partial }}f:=\sum _{j=1}^{n}{\frac {\partial }{\partial {\overline {z}}_{j}}}f{\rm {d}}{\overline {z}}_{j}}$

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 ${\displaystyle f}$ ${\displaystyle {\overline {\partial }}f=0}$

${\displaystyle {\mathrm {d} }f={\overline {\partial }}f+\partial f}$

shown. In the holomorphic case it holds that yes . ${\displaystyle \textstyle \mathrm {d} f=\partial f}$ ${\displaystyle {\overline {\partial }}f=0}$

Dolbeault operators on manifolds

→ Main article : Complex differential form

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 .

Web links

• Eric W. Weisstein : Del Bar Operator . In: MathWorld (English).

literature

• Ingo Lieb & Wolfgang Fischer: Function Theory: Complex Analysis in a Variable , Vieweg & Teubner, 2005, ISBN 978-3-8348-0013-8 .
• Ingo Lieb : The Cauchy-Riemann Complex , Vieweg Aspects of Mathematics, 2002, ISBN 978-3-528-06954-4 .