複素数 z ∈ C z = x + iy C G 上の実可微分関数 f = u + iv : G → C
∂
f
∂
x
=
∂
u
∂
x
+
i
∂
v
∂
x
,
∂
f
∂
y
=
∂
u
∂
y
+
i
∂
v
∂
y
{\displaystyle {\frac {\partial f}{\partial x}}={\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}},\quad {\frac {\partial f}{\partial y}}={\frac {\partial u}{\partial y}}+i{\frac {\partial v}{\partial y}}}
を考えることができる。座標関数として x, y ではなく z = x + iy , z x − iy
可微分関数 f の全微分 df を偏微分 を用いて
d
f
=
∂
f
∂
x
d
x
+
∂
f
∂
y
d
y
{\displaystyle df={\frac {\partial f}{\partial x}}dx+{\frac {\partial f}{\partial y}}dy}
と書くとき、z = x + iy , z x − iy
d
x
=
1
2
(
d
z
+
d
z
¯
)
,
d
y
=
i
2
(
d
z
¯
−
d
z
)
{\displaystyle dx={\frac {1}{2}}(dz+d{\bar {z}}),\quad dy={\frac {i}{2}}(d{\bar {z}}-dz)}
であり、これをもとの全微分に代入して整理したものを形式的に
d
f
=
∂
f
∂
z
d
z
+
∂
f
∂
z
¯
d
z
¯
{\displaystyle df={\frac {\partial f}{\partial z}}dz+{\frac {\partial f}{\partial {\bar {z}}}}d{\bar {z}}}
と書けば、各係数
∂
f
∂
z
:=
1
2
(
∂
f
∂
x
−
i
∂
f
∂
y
)
,
∂
f
∂
z
¯
:=
1
2
(
∂
f
∂
x
+
i
∂
f
∂
y
)
{\displaystyle {\frac {\partial f}{\partial z}}:={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-i{\frac {\partial f}{\partial y}}\right),\qquad {\frac {\partial f}{\partial {\bar {z}}}}:={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}+i{\frac {\partial f}{\partial y}}\right)}
がヴィルティンガー微分 と呼ばれるものである[ 3] ∂f ⁄∂z ∂f ⁄∂z ∂f および ∂ f ∂ コーシー–リーマン作用素 とも呼ばれる。
定義1. 複素平面
C
≡
R
2
=
{
(
x
,
y
)
∣
x
∈
R
,
y
∈
R
}
{\displaystyle \mathbb {C} \equiv \mathbb {R} ^{2}=\{(x,y)\mid x\in \mathbb {R} ,\ y\in \mathbb {R} \}}
線型 偏微分作用素 として定義される：
∂
∂
z
=
1
2
(
∂
∂
x
−
i
∂
∂
y
)
,
∂
∂
z
¯
=
1
2
(
∂
∂
x
+
i
∂
∂
y
)
.
{\displaystyle {\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right),\quad {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}
明らかに、これらの偏微分作用素の自然な定義域 は領域
Ω
⊆
R
2
{\displaystyle \Omega \subseteq \mathbb {R} ^{2}}
C
1
{\displaystyle C^{1}}
線型 であり定数係数 であるから、超関数 の各空間 にただちに拡張できる。
定義2. 複素数体 上のユークリッド空間
C
n
=
R
2
n
=
{
(
x
,
y
)
=
(
x
1
,
…
,
x
n
,
y
1
,
…
,
y
n
)
∣
x
,
y
∈
R
n
}
{\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{2n}=\left\{\left(\mathbf {x} ,\mathbf {y} \right)=\left(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\right)\mid \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}\right\}}
行列 線型 偏微分作用素 として定義される：
{
∂
∂
z
1
=
1
2
(
∂
∂
x
1
−
i
∂
∂
y
1
)
⋮
∂
∂
z
n
=
1
2
(
∂
∂
x
n
−
i
∂
∂
y
n
)
,
{
∂
∂
z
¯
1
=
1
2
(
∂
∂
x
1
+
i
∂
∂
y
1
)
⋮
∂
∂
z
¯
n
=
1
2
(
∂
∂
x
n
+
i
∂
∂
y
n
)
.
{\displaystyle {\begin{cases}{\frac {\partial }{\partial z_{1}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}-i{\frac {\partial }{\partial y_{1}}}\right)\\&\,\vdots \\{\frac {\partial }{\partial z_{n}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}-i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}},\qquad {\begin{cases}{\frac {\partial }{\partial {\bar {z}}_{1}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}+i{\frac {\partial }{\partial y_{1}}}\right)\\&\,\vdots \\{\frac {\partial }{\partial {\bar {z}}_{n}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}+i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}}.}
一変数のときと同様これらの偏微分作用素の自然な定義域は領域
Ω
{\displaystyle \Omega }
2n
C
1
{\displaystyle C^{1}}
この節以降
z
∈
C
n
{\displaystyle z\in \mathbb {C} ^{n}}
複素ベクトル であり
z
≡
(
x
,
y
)
=
(
x
1
,
…
,
x
n
,
y
1
,
…
,
y
n
)
{\displaystyle z\equiv (x,y)=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})}
x
{\displaystyle x}
y
{\displaystyle y}
実ベクトル で n ≥ 1 とする。また、部分集合
Ω
{\displaystyle \Omega }
2n あるいは ℂn 定義1 定義2
補題1.
f
,
g
∈
C
1
(
Ω
)
{\displaystyle f,g\in C^{1}(\Omega )}
α
,
β
{\displaystyle \alpha ,\beta }
複素数 とすると、
i
=
1
,
…
,
n
{\displaystyle i=1,\dots ,n}
∂
∂
z
i
(
α
f
+
β
g
)
=
α
∂
f
∂
z
i
+
β
∂
g
∂
z
i
,
∂
∂
z
¯
i
(
α
f
+
β
g
)
=
α
∂
f
∂
z
¯
i
+
β
∂
g
∂
z
¯
i
{\displaystyle {\frac {\partial }{\partial z_{i}}}\left(\alpha f+\beta g\right)=\alpha {\frac {\partial f}{\partial z_{i}}}+\beta {\frac {\partial g}{\partial z_{i}}},\quad {\frac {\partial }{\partial {\bar {z}}_{i}}}\left(\alpha f+\beta g\right)=\alpha {\frac {\partial f}{\partial {\bar {z}}_{i}}}+\beta {\frac {\partial g}{\partial {\bar {z}}_{i}}}}
補題2.
f
,
g
∈
C
1
(
Ω
)
{\displaystyle f,g\in C^{1}(\Omega )}
i
=
1
,
…
,
n
{\displaystyle i=1,\dots ,n}
積の微分法則 が成り立つ
∂
∂
z
i
(
f
⋅
g
)
=
∂
f
∂
z
i
⋅
g
+
f
⋅
∂
g
∂
z
i
,
∂
∂
z
¯
i
(
f
⋅
g
)
=
∂
f
∂
z
¯
i
⋅
g
+
f
⋅
∂
g
∂
z
¯
i
{\displaystyle {\frac {\partial }{\partial z_{i}}}\left(f\cdot g\right)={\frac {\partial f}{\partial z_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial z_{i}}},\quad {\frac {\partial }{\partial {\bar {z}}_{i}}}\left(f\cdot g\right)={\frac {\partial f}{\partial {\bar {z}}_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial {\bar {z}}_{i}}}}
この性質によってウィルティンガーの微分はちょうど通常の微分のように抽象代数学 的視点の微分 であることに注意。
これは一変数と多変数とで異なる：n > 1 に対して完全な一般性でチェインルール を表現するには2つの領域
Ω
′
⊆
C
m
{\displaystyle \Omega '\subseteq \mathbb {C} ^{m}}
Ω
″
⊆
C
p
{\displaystyle \Omega ''\subseteq \mathbb {C} ^{p}}
g
:
Ω
′
→
Ω
{\displaystyle g:\Omega '\to \Omega }
f
:
Ω
→
Ω
″
{\displaystyle f:\Omega \to \Omega ''}
[ 4]
補題3.1
f
,
g
∈
C
1
(
Ω
)
{\displaystyle f,g\in C^{1}(\Omega )}
g
(
Ω
)
⊆
Ω
{\displaystyle g(\Omega )\subseteq \Omega }
チェインルール が成り立つ
∂
∂
z
(
f
∘
g
)
=
(
∂
f
∂
z
∘
g
)
∂
g
∂
z
+
(
∂
f
∂
z
¯
∘
g
)
∂
g
¯
∂
z
{\displaystyle {\frac {\partial }{\partial z}}\left(f\circ g\right)=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial z}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial z}}}
∂
∂
z
¯
(
f
∘
g
)
=
(
∂
f
∂
z
∘
g
)
∂
g
∂
z
¯
+
(
∂
f
∂
z
¯
∘
g
)
∂
g
¯
∂
z
¯
{\displaystyle {\frac {\partial }{\partial {\bar {z}}}}\left(f\circ g\right)=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial {\bar {z}}}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial {\bar {z}}}}}
補題3.2
g
∈
C
1
(
Ω
′
,
Ω
)
{\displaystyle g\in C^{1}(\Omega ^{\prime },\Omega )}
f
∈
C
1
(
Ω
,
Ω
′
′
)
{\displaystyle \scriptstyle f\in C^{1}(\Omega ,\Omega ^{\prime \prime })}
i
=
1
,
…
,
m
{\displaystyle i=1,\dots ,m}
∂
∂
z
i
(
f
∘
g
)
=
∑
j
=
1
n
(
∂
f
∂
z
j
∘
g
)
∂
g
j
∂
z
i
+
∑
j
=
1
n
(
∂
f
∂
z
¯
j
∘
g
)
∂
g
¯
j
∂
z
i
{\displaystyle {\frac {\partial }{\partial z_{i}}}\left(f\circ g\right)=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial z_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial z_{i}}}}
∂
∂
z
¯
i
(
f
∘
g
)
=
∑
j
=
1
n
(
∂
f
∂
z
j
∘
g
)
∂
g
j
∂
z
¯
i
+
∑
j
=
1
n
(
∂
f
∂
z
¯
j
∘
g
)
∂
g
¯
j
∂
z
¯
i
{\displaystyle {\frac {\partial }{\partial {\bar {z}}_{i}}}\left(f\circ g\right)=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial {\bar {z}}_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial {\bar {z}}_{i}}}}
補題4.
f
∈
C
1
(
Ω
)
{\displaystyle f\in C^{1}(\Omega )}
i
=
1
,
…
,
n
{\displaystyle i=1,\dots ,n}
∂
f
¯
∂
z
i
=
∂
f
¯
∂
z
¯
i
,
∂
f
¯
∂
z
¯
i
=
∂
f
¯
∂
z
i
{\displaystyle {\frac {\overline {\partial f}}{\partial z_{i}}}={\frac {\partial {\bar {f}}}{\partial {\bar {z}}_{i}}},\quad {\frac {\overline {\partial f}}{\partial {\bar {z}}_{i}}}={\frac {\partial {\bar {f}}}{\partial z_{i}}}}
∂
z
∂
z
=
1
,
∂
z
¯
∂
z
=
0
,
∂
z
∂
z
¯
=
0
,
∂
z
¯
∂
z
¯
=
1.
{\displaystyle {\frac {\partial z}{\partial z}}=1,\,{\frac {\partial {\bar {z}}}{\partial z}}=0,\,{\frac {\partial z}{\partial {\bar {z}}}}=0,\,{\frac {\partial {\bar {z}}}{\partial {\bar {z}}}}=1.}
f (z ) が z と z 多項式 であるとき、z , z
∂
∂
z
(
z
3
+
3
z
z
¯
+
z
¯
2
)
=
3
z
2
+
3
z
¯
∂
∂
z
¯
(
z
3
+
3
z
z
¯
+
z
¯
2
)
=
3
z
+
2
z
¯
{\displaystyle {\begin{aligned}{\frac {\partial }{\partial z}}(z^{3}+3z{\bar {z}}+{\bar {z}}^{2})&=3z^{2}+3{\bar {z}}\\{\frac {\partial }{\partial {\bar {z}}}}(z^{3}+3z{\bar {z}}+{\bar {z}}^{2})&=3z+2{\bar {z}}\end{aligned}}}
f が正則 であるとき、f ′ = ∂f である。コーシー・リーマンの方程式 が成り立つことと、∂ f = 0 となることは同値である。∂∂ = ∂ ∂ = (1/4)Δ, ここで Δ = ∂2 /∂x 2 + ∂2 /∂y 2 はラプラシアン 。
正則関数を実部・虚部に分け f = u + iv とすると、Δf = 4∂∂ f = 0, したがって Δu + i Δv = 0 となるから、Δu = Δv = 0, すなわち u と v は調和 であることがわかる。
Andreotti, Aldo (1976) (Italian), Introduzione all'analisi complessa (Lezioni tenute nel febbraio 1972) 24 , Rome: Accademia Nazionale dei Lincei , pp. 34, http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33190 Introduction to complex analysis is a short course in the theory of functions of several complex variables, held on February 1972 at the Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni "Beniamino Segre " .Fichera, Gaetano (1986), “Unification of global and local existence theorems for holomorphic functions of several complex variables”, Memorie della Accademia Nazionale dei Lincei , Classe di Scienze Fisiche, Matematiche e Naturali , 8 18 (3): 61–83, MR 0917525 , Zbl 0705.32006 Gunning, Robert C. ; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables Englewood Cliffs , N.J.: Prentice-Hall , pp. xiv+317, MR 0180696 , Zbl 0141.08601 , https://books.google.co.jp/books?id=L0zJmamx5AAC&printsec=frontcover&hl=it&redir_esc=y#v=onepage&q&f=false Gunning, Robert C. (1990), Introduction to Holomorphic Functions of Several Variables. Volume I: Function Theory , Wadsworth & Brooks/Cole Mathematics Series, Belmont, California : Wadsworth & Brooks/Cole, pp. xx+203, ISBN 0-534-13308-8 , MR 1052649 , Zbl 0699.32001 Henrici, Peter (1993) [1986], Applied and Computational Complex Analysis Volume 3 John Wiley & Sons , pp. X+637, ISBN 0-471-58986-1 , MR 0822470 , Zbl 1107.30300 , https://books.google.it/books?id=vKZPsjaXuF4C&printsec=frontcover#v=onepage&q Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables , North–Holland Mathematical Library, 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland , ISBN 0-444-88446-7 , MR 1045639 , Zbl 0685.32001 Kaup, Ludger; Kaup, Burchard (1983), Holomorphic functions of several variables 3 , Berlin–New York: Walter de Gruyter , pp. XV+349, ISBN 978-3-11-004150-7 , MR 0716497 , Zbl 0528.32001 , https://books.google.co.jp/books?id=nDgBsOurnAIC&printsec=frontcover&redir_esc=y&hl=ja#v=onepage&q Kracht, Manfred; Kreyszig, Erwin (1988), Methods of Complex Analysis in Partial Differential Equations and Applications , Canadian Mathematical Society Series of Monographs and Advanced Texts, New York–Chichester–Brisbane–Toronto–Singapore: John Wiley & Sons , pp. xiv+394, ISBN 0-471-83091-7 , MR 0941372 , Zbl 0644.35005 Martinelli, Enzo (1984) (Italian), Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali 67 , Rome: Accademia Nazionale dei Lincei , pp. 236+II, http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33233 Elementary introduction to the theory of functions of complex variables with particular regard to integral representations " (English translation of the title) are the notes form a course, published by the Accademia Nazionale dei Lincei , held by Martinelli when he was "Professore Linceo ".Remmert, Reinhold (1991), Theory of Complex Functions 122 (Fourth corrected 1998 printing ed.), New York–Berlin–Heidelberg–Barcelona–Hong Kong–London–Milan–Paris–Singapore–Tokyo: Springer Verlag , pp. xx+453, ISBN 0-387-97195-5 , MR 1084167 , Zbl 0780.30001 , https://books.google.co.jp/books?id=CC0dQxtYb6kC&printsec=frontcover&redir_esc=y&hl=ja#v=onepage&q&f=true ISBN 978-0-387-97195-7 . A textbook on complex analysis including many historical notes on the subject.Severi, Francesco (1958) (Italian), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma , Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255, Zbl 0094.28002 Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty . An English translation of the title reads as:-"Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome ".神保, 道夫『複素関数入門』岩波書店 〈現代数学への入門〉、2003年。ISBN 4-00-006874-1 。 野口, 潤次郎『複素解析概論』（第6版）裳華房〈数学選書12〉、2002年。ISBN 978-4-7853-1314-2 。 
