Eine komplexe Zahl
z
∈
C
{\displaystyle z\in \mathbb {C} }
z
:=
x
+
i
y
{\displaystyle z:=x+\mathrm {i} y}
f
:
G
→
C
(
x
,
y
)
→
f
(
x
,
y
)
{\displaystyle f:G\to \mathbb {C} \,{\mbox{ }}\,(x,y)\to f(x,y)}
sei
G
⊂
R
2
{\displaystyle G\subset \mathbb {R} ^{2}}
Gebiet .
Bearbeiten
Die Funktion
f
:
G
→
C
{\displaystyle f:G\to \mathbb {C} }
f
=
u
+
i
v
:
G
→
C
{\displaystyle f=u+\mathrm {i} v\colon G\to \mathbb {C} }
Bearbeiten
Die Realteilfunktion
u
:
G
→
R
(
x
,
y
)
→
u
(
x
,
y
)
{\displaystyle u:G\to \mathbb {R} \,{\mbox{ }}\,(x,y)\to u(x,y)}
und die Imaginärteilfunktion
v
:
G
→
R
(
x
,
y
)
→
v
(
x
,
y
)
{\displaystyle v:G\to \mathbb {R} \,{\mbox{ }}\,(x,y)\to v(x,y)}
sind jeweils eine (reell) differenzierbare Funktion.
Die partiellen Ableitungen von
f
{\displaystyle f}
u
{\displaystyle u}
v
{\displaystyle v}
∂
f
∂
x
(
x
,
y
)
=
∂
u
∂
x
(
x
,
y
)
+
i
∂
v
∂
x
(
x
,
y
)
{\displaystyle {\frac {\partial f}{\partial x}}(x,y)={\frac {\partial u}{\partial x}}(x,y)+\mathrm {i} {\frac {\partial v}{\partial x}}(x,y)}
und
∂
f
∂
y
(
x
,
y
)
=
∂
u
∂
y
(
x
,
y
)
+
i
∂
v
∂
y
(
x
,
y
)
{\displaystyle {\frac {\partial f}{\partial y}}(x,y)={\frac {\partial u}{\partial y}}(x,y)+\mathrm {i} {\frac {\partial v}{\partial y}}(x,y)}
Im nächsten Abschnitt werden nun die Wirtinger-Ableitungen eingeführt, welche ebenfalls partielle Differentialoperatoren sind. Jedoch sind diesWikipedia Authorse einfacher zu berechnen, da die komplexwertige Funktion nicht in Real- und Imaginärteil zerlegt werden muss.
Statt der Koordinaten
x
{\displaystyle x}
y
{\displaystyle y}
z
=
x
+
i
y
{\displaystyle z=x+\mathrm {i} y}
z
¯
=
x
−
i
y
{\displaystyle {\bar {z}}=x-\mathrm {i} y}
Sei
G
⊂
R
2
{\displaystyle G\subset \mathbb {R} ^{2}}
f
:
G
→
C
{\displaystyle f:G\rightarrow \mathbb {C} }
Umgebung
U
⊆
G
{\displaystyle U\subseteq G}
z
o
{\displaystyle z_{o}}
Richtungsableitung von
f
{\displaystyle f}
z
o
=
(
x
o
,
y
o
)
∈
R
2
{\displaystyle z_{o}=(x_{o},y_{o})\in \mathbb {R} ^{2}}
in Richtung des Vektors
v
=
(
v
1
,
v
2
)
∈
R
2
{\displaystyle v=(v_{1},v_{2})\in \mathbb {R} ^{2}}
∂
f
∂
v
(
z
o
)
:=
lim
h
→
0
f
(
z
o
+
h
⋅
v
)
−
f
(
z
o
)
h
{\displaystyle {\frac {\partial f}{\partial v}}(z_{o}):=\lim _{h\to 0}{\frac {f(z_{o}+h\cdot v)-f(z_{o})}{h}}}
Der Limes muss existieren.
Partielle Ableitung kann man as spezielle Ableitungen mit
v
=
(
1
,
0
)
{\displaystyle v=(1,0)}
v
=
(
0
,
1
)
{\displaystyle v=(0,1)}
Bei einer total differenzierbaren Funktion
f
:
G
→
C
{\displaystyle f:G\rightarrow \mathbb {C} }
⟨
v
,
G
r
a
d
(
f
)
(
z
o
)
⟩
{\displaystyle \langle v,Grad(f)(z_{o})\rangle }
v
{\displaystyle v}
Gradient von
f
{\displaystyle f}
z
o
{\displaystyle z_{o}}
Komplexe Konjugation und Richtungsableitungen
Bearbeiten
Aus den komplexen Zahlen
z
=
x
+
i
y
{\displaystyle z=x+\mathrm {i} y}
z
¯
=
x
−
i
y
{\displaystyle {\bar {z}}=x-\mathrm {i} y}
R
2
{\displaystyle \mathbb {R} ^{2}}
z
=
(
x
,
y
)
{\displaystyle z=(x,y)}
z
¯
=
(
x
,
−
y
)
{\displaystyle {\overline {z}}=(x,-y)}
∂
f
∂
x
(
z
o
)
{\displaystyle {\frac {\partial f}{\partial x}}(z_{o})}
∂
f
∂
y
(
z
o
)
{\displaystyle {\frac {\partial f}{\partial y}}(z_{o})}
f
{\displaystyle f}
Eine Funktion
f
{\displaystyle f}
x
o
+
i
y
o
{\displaystyle x_{o}+iy_{o}}
u
,
v
{\displaystyle u,v}
u
:
G
→
R
{\displaystyle u:G\rightarrow \mathbb {R} }
v
:
G
→
R
{\displaystyle v:G\rightarrow \mathbb {R} }
G
⊂
R
2
{\displaystyle G\subset \mathbb {R} ^{2}}
z
o
=
(
x
o
,
y
o
)
∈
R
2
{\displaystyle z_{o}=(x_{o},y_{o})\in \mathbb {R} ^{2}}
∂
u
∂
x
(
x
o
,
y
o
)
=
∂
v
∂
y
(
x
o
,
y
o
)
{\displaystyle {\frac {\partial u}{\partial x}}(x_{o},y_{o})={\frac {\partial v}{\partial y}}(x_{o},y_{o})}
∂
u
∂
y
(
x
o
,
y
o
)
=
−
∂
v
∂
x
(
x
o
,
y
o
)
{\displaystyle \displaystyle {\frac {\partial u}{\partial y}}(x_{o},y_{o})=-{\frac {\partial v}{\partial x}}(x_{o},y_{o})}
erfüllt sind und
∂
f
∂
x
(
z
o
)
=
∂
u
∂
x
(
z
o
)
+
i
⋅
∂
v
∂
x
(
z
o
)
{\displaystyle {\frac {\partial f}{\partial x}}(z_{o})={\frac {\partial u}{\partial x}}(z_{o})+i\cdot {\frac {\partial v}{\partial x}}(z_{o})}
∂
f
∂
y
(
z
o
)
=
∂
u
∂
y
(
z
o
)
+
i
⋅
∂
v
∂
y
(
z
o
)
{\displaystyle {\frac {\partial f}{\partial y}}(z_{o})={\frac {\partial u}{\partial y}}(z_{o})+i\cdot {\frac {\partial v}{\partial y}}(z_{o})}
Richtungsableitungen und partielle Ableitungen
Bearbeiten
Definiert man mit CRG die so genannten Wirtinger-Ableitungen
∂
f
∂
z
(
z
o
)
:=
1
2
⋅
(
∂
f
∂
x
(
z
o
)
−
i
⋅
∂
f
∂
y
(
z
o
)
)
{\displaystyle {\frac {\partial f}{\partial z}}(z_{o}):={\frac {1}{2}}\cdot \left({\frac {\partial f}{\partial x}}(z_{o})-i\cdot {\frac {\partial f}{\partial y}}(z_{o})\right)}
und
∂
f
∂
z
¯
(
z
o
)
:=
1
2
⋅
(
∂
f
∂
x
(
z
o
)
+
i
⋅
∂
f
∂
y
(
z
o
)
)
,
{\displaystyle {\frac {\partial f}{\partial {\overline {z}}}}(z_{o}):={\frac {1}{2}}\cdot \left({\frac {\partial f}{\partial x}}(z_{o})+i\cdot {\frac {\partial f}{\partial y}}(z_{o})\right),}
so erhält man
∂
f
∂
z
¯
(
z
o
)
=
0
{\displaystyle {\frac {\partial f}{\partial {\overline {z}}}}(z_{o})=0}
Holomorphiekriterium .
Durch das Einsetzen der Wirtinger-Ableitungen erhält man jeweils die partielle Ableitungen mit:
1
2
⋅
(
∂
f
∂
z
(
z
o
)
+
∂
f
∂
z
¯
(
z
o
)
)
=
1
2
⋅
(
(
∂
f
∂
x
(
z
o
)
−
i
⋅
∂
f
∂
x
(
z
o
)
)
+
(
∂
f
∂
x
(
z
o
)
+
i
⋅
∂
f
∂
y
(
z
o
)
)
)
=
1
2
⋅
(
2
⋅
∂
f
∂
x
(
z
o
)
)
=
∂
f
∂
x
(
z
o
)
i
2
⋅
(
∂
f
∂
z
(
z
o
)
−
∂
f
∂
z
¯
(
z
o
)
)
=
i
2
⋅
(
(
∂
f
∂
x
(
z
o
)
−
i
⋅
∂
f
∂
x
(
z
o
)
)
−
(
∂
f
∂
x
(
z
o
)
+
i
⋅
∂
f
∂
y
(
z
o
)
)
)
=
i
2
⋅
(
−
2
i
⋅
∂
f
∂
y
(
z
o
)
)
=
∂
f
∂
y
(
z
o
)
{\displaystyle {\begin{array}{rcl}{\frac {1}{2}}\cdot \left({\frac {\partial f}{\partial z}}(z_{o})+{\frac {\partial f}{\partial {\overline {z}}}}(z_{o})\right)&=&{\frac {1}{2}}\cdot \left(\left({\frac {\partial f}{\partial x}}(z_{o})-i\cdot {\frac {\partial f}{\partial x}}(z_{o})\right)+\left({\frac {\partial f}{\partial x}}(z_{o})+i\cdot {\frac {\partial f}{\partial y}}(z_{o})\right)\right)\\&=&{\frac {1}{2}}\cdot \left(2\cdot {\frac {\partial f}{\partial x}}(z_{o})\right)={\frac {\partial f}{\partial x}}(z_{o})\\{\frac {i}{2}}\cdot \left({\frac {\partial f}{\partial z}}(z_{o})-{\frac {\partial f}{\partial {\overline {z}}}}(z_{o})\right)&=&{\frac {i}{2}}\cdot \left(\left({\frac {\partial f}{\partial x}}(z_{o})-i\cdot {\frac {\partial f}{\partial x}}(z_{o})\right)-\left({\frac {\partial f}{\partial x}}(z_{o})+i\cdot {\frac {\partial f}{\partial y}}(z_{o})\right)\right)\\&=&{\frac {i}{2}}\cdot \left(-2i\cdot {\frac {\partial f}{\partial y}}(z_{o})\right)={\frac {\partial f}{\partial y}}(z_{o})\end{array}}}
Bearbeiten
Insgesamt erhält man mit
z
=
x
+
i
y
{\displaystyle z=x+iy}
C
{\displaystyle \mathbb {C} }
x
=
1
2
(
z
+
z
¯
)
{\displaystyle \textstyle x={\frac {1}{2}}(z+{\bar {z}})}
y
=
1
2
i
(
z
−
z
¯
)
=
i
2
(
z
¯
−
z
)
{\displaystyle \textstyle y={\frac {1}{2\mathrm {i} }}(z-{\bar {z}})={\frac {\mathrm {i} }{2}}({\bar {z}}-z)}
R
2
{\displaystyle \mathbb {R} ^{2}}
G
⊂
R
2
{\displaystyle G\subset \mathbb {R} ^{2}}
f
:
G
→
C
{\displaystyle f:G\rightarrow \mathbb {C} }
∂
f
∂
x
(
z
o
)
=
1
2
⋅
(
∂
f
∂
z
(
z
o
)
+
∂
f
∂
z
¯
(
z
o
)
)
∈
C
{\displaystyle {\frac {\partial f}{\partial x}}(z_{o})={\frac {1}{2}}\cdot \left({\frac {\partial f}{\partial z}}(z_{o})+{\frac {\partial f}{\partial {\overline {z}}}}(z_{o})\right)\in \mathbb {C} }
und
∂
f
∂
y
(
z
o
)
=
i
2
⋅
(
∂
f
∂
z
¯
(
z
o
)
−
∂
f
∂
z
(
z
o
)
)
∈
C
{\displaystyle {\frac {\partial f}{\partial y}}(z_{o})={\frac {i}{2}}\cdot \left({\frac {\partial f}{\partial {\overline {z}}}}(z_{o})-{\frac {\partial f}{\partial z}}(z_{o})\right)\in \mathbb {C} }
Für die Differentiale erhält man daraus
d
x
=
1
2
(
d
z
+
d
z
¯
)
{\displaystyle \mathrm {d} x={\frac {1}{2}}(\mathrm {d} z+\mathrm {d} {\bar {z}})}
d
y
=
i
2
(
d
z
¯
−
d
z
)
{\displaystyle \mathrm {d} y={\frac {\mathrm {i} }{2}}(\mathrm {d} {\bar {z}}-\mathrm {d} z)}
Einsetzen in das totale Differential und Umsortieren liefert
d
f
=
1
2
(
∂
f
∂
x
−
i
∂
f
∂
y
)
d
z
+
1
2
(
∂
f
∂
x
+
i
∂
f
∂
y
)
d
z
¯
{\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}}}
Um (formal) die Beziehung
d
f
=
∂
f
∂
z
d
z
+
∂
f
∂
z
¯
d
z
¯
{\displaystyle \mathrm {d} f={\frac {\partial f}{\partial z}}\mathrm {d} z+{\frac {\partial f}{\partial {\bar {z}}}}\mathrm {d} {\bar {z}}}
∂
f
∂
z
:=
1
2
(
∂
f
∂
x
−
i
∂
f
∂
y
)
{\displaystyle {\frac {\partial f}{\partial z}}:={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-\mathrm {i} {\frac {\partial f}{\partial y}}\right)}
und
∂
f
∂
z
¯
:=
1
2
(
∂
f
∂
x
+
i
∂
f
∂
y
)
{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}:={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}+\mathrm {i} {\frac {\partial f}{\partial y}}\right)}
Dies sind die Wirtinger-Ableitungen .
Für
∂
f
∂
z
{\displaystyle \textstyle {\frac {\partial f}{\partial z}}}
∂
f
{\displaystyle \,\partial f}
∂
f
∂
z
¯
{\displaystyle \textstyle {\frac {\partial f}{\partial {\bar {z}}}}}
∂
¯
f
{\displaystyle {\bar {\partial }}f}
∂
¯
{\displaystyle {\overline {\partial }}}
Cauchy-Riemann-Operator .
Mit Hilfe der partiellen Ableitungen schreibt sich das (totale) Differential von
f
{\displaystyle f}
d
f
=
∂
f
∂
x
d
x
+
∂
f
∂
y
d
y
{\displaystyle \mathrm {d} f={\frac {\partial f}{\partial x}}\mathrm {d} x+{\frac {\partial f}{\partial y}}\mathrm {d} y}
als Richtungsableitungen für
v
=
(
1
,
1
)
{\displaystyle v=(1,1)}
v
¯
=
(
1
,
−
1
)
{\displaystyle {\overline {v}}=(1,-1)}
Der Wirtinger-Kalkül findet insbesondere in der Funktionentheorie Anwendung, da für holomorphe Funktionen die Notation sich auf ein Minimum reduziert. Außerdem ist dieser Kalkül sehr stabil, wie Eigenschaften 3 und 4 im nächsten Abschnitt zeigen.
Eine reell differenzierbare Funktion ist genau dann eine holomorphe Funktion , wenn
∂
¯
f
=
0
{\displaystyle {\overline {\partial }}f=0}
∂
f
{\displaystyle \partial f}
f
{\displaystyle f}
∂
¯
f
=
0
{\displaystyle {\overline {\partial }}f=0}
Cauchy-Riemannschen Differentialgleichungen ist. Aus diesem Grund trägt der Operator
∂
¯
{\displaystyle {\overline {\partial }}}
Gilt hingegen für eine reell differenzierbare Funktion
f
{\displaystyle f}
∂
f
=
0
{\displaystyle \partial f=0}
antiholomorph und das reelle Differential kann mit Hilfe von Eigenschaft 1 aus
∂
¯
f
{\displaystyle {\overline {\partial }}f}
Es gelten die Gleichungen
∂
f
∂
x
=
∂
f
+
∂
¯
f
{\displaystyle {\frac {\partial f}{\partial x}}=\partial f+{\overline {\partial }}f}
und
∂
f
∂
y
=
i
(
∂
f
−
∂
¯
f
)
{\displaystyle {\frac {\partial f}{\partial y}}=\mathrm {i} \left(\partial f-{\overline {\partial }}f\right)}
Die Operatoren
∂
{\displaystyle \partial }
∂
¯
{\displaystyle {\overline {\partial }}}
C
{\displaystyle \mathbb {C} }
linear , das heißt für
a
,
b
∈
C
{\displaystyle a,b\in \mathbb {C} }
f
,
g
:
G
→
C
{\displaystyle f,g\colon G\to \mathbb {C} }
∂
(
a
f
+
b
g
)
=
a
∂
f
+
b
∂
g
{\displaystyle \partial (af+bg)=a\partial f+b\partial g}
und
∂
¯
(
a
f
+
b
g
)
=
a
∂
¯
f
+
b
∂
¯
g
{\displaystyle {\overline {\partial }}(af+bg)=a{\overline {\partial }}f+b{\overline {\partial }}g}
Für jede reell differenzierbare Funktion
f
{\displaystyle f}
∂
¯
f
=
∂
f
¯
¯
{\displaystyle {\overline {\partial }}f={\overline {\partial {\overline {f}}}}}
und
∂
¯
f
¯
=
∂
f
¯
{\displaystyle {\overline {\partial }}\ {\overline {f}}={\overline {\partial f}}}
Für die Wirtinger-Ableitungen gilt die Kettenregel
∂
(
g
∘
f
)
∂
z
(
z
0
)
=
∂
g
∂
w
(
f
(
z
0
)
)
⋅
∂
f
∂
z
(
z
0
)
+
∂
g
∂
w
¯
(
f
(
z
0
)
)
⋅
∂
f
¯
∂
z
(
z
0
)
{\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})}
und
∂
(
g
∘
f
)
∂
z
¯
(
z
0
)
=
∂
g
∂
w
(
f
(
z
0
)
)
⋅
∂
f
∂
z
¯
(
z
0
)
+
∂
g
∂
w
¯
(
f
(
z
0
)
)
⋅
∂
f
¯
∂
z
¯
(
z
0
)
{\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})}
Das Hauptsymbol von
∂
{\displaystyle \partial }
ξ
↦
1
2
(
ξ
1
−
i
ξ
2
)
{\displaystyle \xi \mapsto {\tfrac {1}{2}}(\xi _{1}-\mathrm {i} \xi _{2})}
∂
¯
{\displaystyle {\overline {\partial }}}
ξ
↦
1
2
(
ξ
1
+
i
ξ
2
)
{\displaystyle \xi \mapsto {\tfrac {1}{2}}(\xi _{1}+\mathrm {i} \xi _{2})}
elliptisch .
Assoziierter Laplace- und Dirac-Operator
Bearbeiten
Mit den Wirtinger-Ableitungen kann man den Laplace-Operator durch
Δ
f
=
4
∂
∂
¯
f
=
4
∂
¯
∂
f
{\displaystyle \Delta f=4\partial {\overline {\partial }}f=4{\overline {\partial }}\partial f}
darstellen. Daraus folgt insbesondere, dass der Operator
D
:=
2
(
0
−
∂
∂
¯
0
)
{\displaystyle D:=2{\begin{pmatrix}0&-\partial \\{\overline {\partial }}&0\end{pmatrix}}}
ein Dirac-Operator ist.
Die Fundamentallösung des Cauchy-Riemann-Operators
∂
∂
z
¯
{\displaystyle \textstyle {\frac {\partial }{\partial {\overline {z}}}}}
1
π
z
{\displaystyle \textstyle {\frac {1}{\pi z}}}
u
(
z
)
=
1
π
z
{\displaystyle \textstyle u(z)={\frac {1}{\pi z}}}
Distribution löst die Gleichung
∂
∂
z
¯
u
(
z
)
=
δ
{\displaystyle \textstyle {\frac {\partial }{\partial {\overline {z}}}}u(z)=\delta }
δ
{\displaystyle \delta }
Delta-Distribution ist. Eine Herleitung ist im Artikel Cauchy-Riemannsche partielle Differentialgleichungen zu finden.
