ウィルティンガーの微分のソースを表示

[[file:Wilhelm Wirtinger.jpg|thumb|right|ヴィルヘルム・ヴィルティンガー]]
[[複素解析|一変数]]および[[複素多変数|多変数の複素解析]]において、'''ウィルティンガーの微分'''（{{lang-en-short|Wirtinger derivative}}, ときに {{en|'''Wirtinger operator'''}}<ref>See references {{Harvnb|Fichera|1986|p=62}} and {{harvnb|Kracht|Kreyszig|1988|p=10}}.</ref> とも）は、複素多変数関数論に関する研究において1927年に導入した{{仮リンク|ヴィルヘルム・ヴィルティンガー|en|Wilhelm Wirtinger}} (Wilhelm Wirtinger) の名前にちなんでおり、[[正則関数]]、{{仮リンク|反正則関数|en|antiholomorphic function}}、あるいは単に[[領域 (解析学)|複素領域]]上の[[微分可能な関数]]に適用したときに、1つの{{仮リンク|実変数|en|Function of a real variable}}に関して通常の[[微分]]と非常によく似た振る舞いをする、一階の[[偏微分作用素]]である。これらの作用素によって、そのような関数に対する[[微分学]]の、{{仮リンク|実変数関数|en|Function of a real variable}}に対する通常の微分学と完全に類似した、構成ができる<ref>ウィルティンガーの微分の基本的な性質のいくつかは、常（あるいは偏）[[導関数|微分]]を特徴づけ、通常の微分学の構成するために使われるのと同じものである。</ref>。

== 導入 ==
複素数 {{math|''z'' &isin; '''C'''}} を実部と虚部に分解して {{math|''z'' {{=}} ''x'' + ''iy''}} と書き、{{math|'''C'''}} の適当な領域 {{mvar|G}} 上の実可微分関数 {{math|''f'' {{=}} ''u'' + ''iv'': ''G'' &rarr; '''C'''}} に対し、偏微分
:<math>\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}</math>
を考えることができる。座標関数として {{mvar|x, y}} ではなく {{math|''z'' {{=}} ''x'' + ''iy'', {{overline|''z''}} {{=}} ''x'' &minus; ''iy''}} を考えるとき、これとは別の偏微分作用素としてヴィルティンガー微分が定義されるが、複素数値関数を実部と虚部に明示的に分けずとも計算できるため扱いはより平易なものとなる。 

可微分関数 {{mvar|f}} の[[全微分]] {{mvar|df}} を[[偏微分]]を用いて
: <math>df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy</math>
と書くとき、{{math|''z'' {{=}} ''x'' + ''iy'', {{overline|''z''}} {{=}} ''x'' &minus; ''iy''}} とすれば微分小に関して
: <math>dx = \frac{1}{2}(dz + d\bar z),\quad dy =  \frac{i}{2}(d\bar z -dz)</math>
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 
:<math>df = \frac{\partial f}{\partial z} dz + \frac{\partial f}{\partial \bar{z}} d\bar z</math>
と書けば、各係数
:<math>\frac{\partial f}{\partial z}:= \frac12\left(\frac{\partial f}{\partial x}-i\frac{\partial f}{\partial y}\right),\qquad \frac{\partial f}{\partial \bar{z}}:= \frac12\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\right)</math>
が'''ヴィルティンガー微分'''と呼ばれるものである<ref>もちろん記号 {{math|∂}} は[[偏微分]]を意味するものではないが、ヴィルティンガー微分は、{{mvar|z}} と {{math|{{overline|''z''}}}} を独立変数であるかのように扱えば、[[偏微分#定義|通常の変数変換の公式]]を形式的に適用したものと同一であるし、後述のように偏微分と同様の性質も持つ。</ref>。しばしば {{math|{{fraction|&part;''f''|&part;''z''}}}} および {{math|{{fraction|&part;''f''|&part;{{overline|''z''}}}}}} をそれぞれ {{math|&part;''f''}} および {{math|{{overline|&part;}}''f''}} とも書き、また作用素 {{math|{{overline|&part;}}}} は'''コーシー&ndash;リーマン作用素'''とも呼ばれる。

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==Historical notes==

===Early days (1899–1911): the work of Henri Poincaré===

Wirtinger derivatives were used in [[complex analysis]] at least as early as in the paper {{Harv|Poincaré|1899}}, as briefly noted by {{Harvtxt|Cherry|Ye|2001|p=31}} and by {{Harvtxt|Remmert|1991|pp=66–67}}.<ref>Reference to the work {{Harvnb|Poincaré|1899}} of [[Henri Poincaré]] is precisely stated by {{Harvtxt|Cherry|Ye|2001}}, while [[Reinhold Remmert]] does not cite any reference to support his assertion.</ref> As a matter of fact, in the third paragraph of his 1899 paper,<ref>See reference {{Harv|Poincaré|1899|pp=111–114}}</ref> [[Henri Poincaré]] first defines the [[complex variable]] in ℂ<sup>''n''</sup> and its [[complex conjugate]] as follows

:<math>x_k+iy_k=z_k\qquad x_k-iy_k=u_k</math>

where the index <math>k</math> ranges from 1 to <math>n</math>. Then he writes the equation defining the functions <math>V</math> he calls ''biharmonique'',<ref>These functions are precisely [[pluriharmonic function]]s, and the [[linear differential operator]] defining them, i.e. the operator in equation 2 of {{Harv|Poincaré|1899|p=112}}, is exactly the [[Dimension|''n''-dimensional]] [[pluriharmonic operator]].</ref> previously written using [[partial derivative]]s with respect to the [[Real number|real]] [[Variable (mathematics)|variables]] <math>x_k</math>, <math>y_q</math> with <math>k</math>,  <math>q</math> ranging from 1 to <math>n</math>, exactly in the following way<ref>See {{Harv|Poincaré|1899|p=112}}, equation 2': note that, throughout the paper, the symbol <math>d</math> is used to signify [[Partial derivative|partial differentiation]] respect to a given [[Variable (mathematics)|variable]], instead of the now commonplace symbol&nbsp;∂.</ref>

:<math>\frac{d^2 V}{dz_k \, du_q}=0</math>

This implies that he implicitly used {{EquationNote|2|definition 2}} below: to see this is sufficient to compare equations 2 and 2' of {{Harv|Poincaré|1899|p=112}}. Apparently, this paper was not noticed by the early savants doing research in the [[Several complex variables|theory of functions of several complex variables]]: in the papers of {{Harvtxt|Levi-Civita|1905}}, {{Harvtxt|Levi|1910}} (and {{Harvnb|Levi|1911}}) and of {{Harvtxt|Amoroso|1912}} all fundamental [[partial differential operator]]s of the theory are expressed directly by using [[partial derivatives]] respect to the [[real part|real]] and [[imaginary part]]s of the [[complex variable]]s involved. In the long survey paper by {{Harvtxt|Osgood|1966}} (first published in 1913),<ref>The corrected [[Dover Publications|Dover edition]] of the paper {{Harv|Osgood|1913}} contains much important historical information on the early development of the [[Several complex variables|theory of functions of several complex variables]], and is therefore a useful source.</ref> [[partial derivatives]] with respect to each [[complex variable]] of a [[Several complex variables|holomorphic function of several complex variables]] seem to be meant as [[formal derivative]]s: as a matter of fact when [[William Fogg Osgood|Osgood]] express the [[pluriharmonic operator]]<ref>See {{Harvtxt|Osgood|1966|pp=23–24}}: curiously, he calls ''[[Cauchy–Riemann equations]]'' this set of equations.</ref> and the [[Levi operator]], he follows the established practice of [[Luigi Amoroso|Amoroso]], [[Eugenio Elia Levi|Levi]] and [[Tullio Levi-Civita|Levi-Civita]].

===The work of Dimitrie Pompeiu in 1912 and 1913: a new formulation===

According to {{Harvtxt|Henrici|1993|p=294}}, a new step in the definition of the concept was taken by [[Dimitrie Pompeiu]]: in the paper {{Harv|Pompeiu|1912}}, given a [[Complex number|complex valued]] [[differentiable function]] (in the sense of [[real analysis]]) of one [[complex variable]] <math>g(z)</math> defined in the [[Neighbourhood (mathematics)|neighbourhood]] of a given [[Point (mathematics)|point]] <math>z_0</math>∈ℂ, he defines the [[areolar derivative]] as the following [[Limit (mathematics)|limit]]

:<math>{\frac{\partial g}{\partial \bar{z}}(z_0)}\overset{\mathrm{def}}{=}\lim_{r \to 0}\frac{1}{2\pi i r^2}{\oint_{\Gamma(z_0,r)}\!\!\!\!\!\!\!\!\!\!g(z){\mathrm{d}z}}</math>

where <math>\scriptstyle \Gamma_{(z_0,r)}=\partial D(z_0,r)</math> is the [[Boundary (topology)|boundary]] of a [[Disk (mathematics)|disk]] of radius <math>r</math> entirely contained in the [[Domain of a function|domain of definition]] of <math>g(z)</math>, i.e. his bounding [[circle]].<ref>This is the definition given by {{Harvtxt|Henrici|1993|p=294|}} in his approach to [[Dimitrie Pompeiu|Pompeiu's work]]: as {{Harvtxt|Fichera|1969|p=27}} remarks, the original definition of {{Harvtxt|Pompeiu|1912}} does not require the [[Domain (mathematical analysis)|domain]] of [[Integration (mathematics)|integration]] to be a [[circle]]. See the entry [[areolar derivative]] for further information.</ref> This is evidently an alternative definition of Wirtinger derivative respect to the [[complex conjugate]] [[Variable (mathematics)|variable]]:<ref>See the section "[[Wirtinger derivatives#Formal definition|Formal definition]]" of this entry.</ref> it is a more general one, since, as noted a by {{Harvtxt|Henrici|1993|p=294}}, the limit may exists for functions that are not even [[Differentiable function|differentiable]] at <math>z=z_0</math>.<ref>See problem 2 in {{Harvnb|Henrici|1993|p=294}} for one example of such a function.</ref> According to {{Harvtxt|Fichera|1969|p=28}}, the first to identify the [[areolar derivative]] as a [[weak derivative]] in the [[Generalized derivative|sense of Sobolev]] was [[Ilia Vekua]].<ref>See also the excellent book by {{Harvtxt|Vekua|1962|p=55}}, theorem 1.31: ''If the generalized derivative <math>\scriptstyle\partial_{\bar{z}}w</math>∈'''[[Lp space|<math>L_p(\Omega)</math>]]''', p>1, then the function <math>w(z)</math> has [[almost everywhere]] in <math>G</math> a derivative in the sense of [[Dimitrie Pompeiu|Pompeiu]], the latter being equal to the [[Generalized derivative]] in the sense of [[Sergei Sobolev|Sobolev]] <math>\scriptstyle\partial_{\bar{z}}w</math>''.</ref> In his following paper, {{Harvtxt|Pompeiu|1913}} uses this newly defined concept in order to introduce his generalization of [[Cauchy's integral formula]], the now called [[Cauchy–Pompeiu formula]].

===The work of Wilhelm Wirtinger===

The first systematic introduction of Wirtinger derivatives seems due to [[Wilhelm Wirtinger]] in the paper {{Harvnb|Wirtinger|1926}} in order to simplify the calculations of quantities occurring in the [[Several complex variables|theory of functions of several complex variables]]: as a result of the introduction of these [[differential operator]]s, the form of all the differential operators commonly used in the theory, like the [[Levi operator]] and the [[Cauchy–Riemann equations|Cauchy–Riemann operator]], is considerably simplified and consequently easier to handle. The paper is deliberately written from a formal point of view, i.e. without giving a rigorous derivation of the properties deduced.
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==定義==
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Despite their ubiquitous use,<ref>With or without the attribution of the concept to [[Wilhelm Wirtinger]]: see, for example, the well known monograph {{harvnb|Hörmander|1990|p=1,23}}.</ref> it seems that there is no text listing all the properties of Wirtinger derivatives: however, fairly complete references are the short course on [[Several complex variables|multidimensional complex analysis]] by {{Harvtxt|Andreotti|1976|pp=3–5}},<ref>In this course lectures, [[Aldo Andreotti]] uses the properties of Wirtinger derivatives in order to prove the [[Closure (mathematics)|closure]] of the [[Algebra over a field|algebra]] of [[holomorphic function]]s under certain [[Operation (mathematics)|operations]]: this purpose is common to all references cited in this section.</ref> the [[monograph]] of {{Harvtxt|Gunning|Rossi|1965|pp=3–6}},<ref>This is a classical work on the [[Several complex variables|theory of functions of several complex variables]] dealing mainly with its [[Sheaf theory|sheaf theoretic]] aspects: however, in the introductory sections, Wirtinger derivatives and a few other analytical tools are introduced and their application to the theory is described.</ref> and the monograph of {{Harvtxt|Kaup|Kaup|1983|p=2,4}}<ref>In this work, the authors prove some of the properties of Wirtinger derivatives also for the general case of <math>C^1</math> [[Function (mathematics)|functions]]: in this single aspect, their approach is different from the one adopted by the other authors cited in this section, and perhaps more complete.</ref> which are used as general references in this and the following sections.
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===一変数の場合===
{{EquationRef|1|定義1.}} [[複素平面]] <math>\mathbb C \equiv \mathbb R^2 = \{(x,y) \mid x\in \mathbb R,\  y\in\mathbb R\} </math> を考えよう。ウィルティンガーの微分は次の一階[[線型作用素|線型]][[偏微分作用素]]として定義される：

:<math> \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).</math>

明らかに、これらの偏微分作用素の自然な[[定義域]]は[[領域 (解析学)|領域]] <math>\Omega \subseteq \mathbb R^2</math> 上の [[滑らかな関数|<math>C^1</math> 級関数]]の空間であるが、これらの作用素は[[線型]]であり[[定数係数]]であるから、[[超関数]]の各[[関数空間|空間]]にただちに拡張できる。

===多変数の場合===
{{EquationRef|2|定義2.}} [[複素数体]]上の[[ユークリッド空間]] <math>\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\}</math> を考えよう。ウィルティンガーの微分は次の一階[[行列 (数学)|行列]][[線型作用素|線型]][[偏微分作用素]]として定義される：

:<math>\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}.</math>
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 '''<math>\Omega</math>'''&nbsp;⊆&nbsp;ℝ<sup>''2n''</sup> 上の <math>C^1</math> 級関数の空間であるが定数係数の線型作用素のため超関数の空間へと拡張できる。

== 基本的な性質 ==
この節以降 <math>z \in \mathbb C^n</math> は[[複素ベクトル]]であり <math>z \equiv (x,y) = (x_1,\ldots,x_n,y_1,\ldots,y_n)</math> ただし <math>x</math>,&nbsp;<math>y</math> は[[実ベクトル]]で ''n''&nbsp;≥&nbsp;1 とする。また、[[部分集合]] '''<math>\Omega</math>''' は ℝ<sup>2''n''</sup> あるいは ℂ<sup>''n''</sup> の領域とする。証明は全て{{EquationNote|1|定義1}}、{{EquationNote|2|定義2}}、そして（常あるいは偏）微分の対応する性質の容易な結果である。

=== 線型性 ===
{{EquationRef|3|補題1.}} <math>f,g \in C^1(\Omega)</math> とし、<math>\alpha,\beta</math> を[[複素数]]とすると、<math>i=1,\dots,n</math> に対して、以下の等式が成り立つ

:<math> \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}</math>

=== 積の法則 ===
{{EquationRef|4|補題2.}} <math>f,g \in C^1(\Omega)</math> であれば、<math>i= 1,\dots,n</math> に対して、[[積の微分法則]]が成り立つ

:<math> \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}</math>

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[[導分 (抽象代数学)|微分]]であることに注意。

===チェインルール===
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[[連鎖律|チェインルール]]を表現するには2つの領域 <math>\Omega'\subseteq\mathbb C^m</math> および <math>\Omega''\subseteq\mathbb C^p</math> と自然な滑らかさの要求を満たす2つの関数 <math>g: \Omega'\to\Omega </math> および <math>f:\Omega \to \Omega''</math> を考える必要がある<ref>See {{harvnb|Kaup|Kaup|1983|p=4}} and also {{harvnb|Gunning|1990|p=5}}: [[Robert Gunning (mathematician)|Gunning]] considers the general case of [[Smooth function#Differentiability classes|<math>C^1</math> functions]] but only for ''p''&nbsp;=&nbsp;1. References {{harvnb|Andreotti|1976|p=5}} and {{harvnb|Gunning|Rossi|1965|p=6}}, as already pointed out, consider only [[holomorphic function|holomorphic maps]] with ''p''&nbsp;=&nbsp;1: however, the resulting formulas are formally very similar.</ref>。

====一変数の場合====
{{EquationRef|5|補題3.1}} <math>f,g \in C^1(\Omega)</math> および <math>g(\Omega) \subseteq \Omega</math> であれば、[[連鎖律|チェインルール]]が成り立つ

:<math> \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}
</math>
:<math> \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}}</math>

==== 多変数の場合====
{{EquationRef|6|補題3.2}} <math>g \in C^1(\Omega^\prime,\Omega)</math> および <math>\scriptstyle f \in C^1(\Omega,\Omega^{\prime\prime})</math> であれば、<math>i= 1,\dots,m</math> に対し、以下の形のチェインルールが成り立つ

:<math> \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}</math>
:<math>\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}</math>

===共役===
{{EquationRef|7|補題4.}} <math>f\in C^1(\Omega)</math> であれば、<math>i=1,\dots,n</math> に対して、以下の等式が成り立つ

:<math>\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}</math>

== 例 ==
* <math>\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.</math>
* ''f''(''z'') が ''z'' と {{overline|''z''}} の[[多項式]]であるとき、''z'', {{overline|''z''}} を独立変数と思って形式的に偏微分すればよい。例えば、
::<math>\begin{align}
\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{align}</math>
* ''f'' が[[正則関数|正則]]であるとき、''f''&thinsp;&prime; = ∂''f'' である。
* [[コーシー・リーマンの方程式]]が成り立つことと、{{overline|∂}}''f'' = 0 となることは同値である。
* ∂{{overline|∂}} = {{overline|∂}}∂ = (1/4)Δ, ここで Δ = ∂<sup>2</sup>/∂''x''<sup>2</sup> + ∂<sup>2</sup>/∂''y''<sup>2</sup> 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、Δ''f'' = 4∂{{overline|∂}}''f'' = 0, したがって Δ''u'' + ''i''Δ''v'' = 0 となるから、Δ''u'' = Δ''v'' = 0, すなわち ''u'' と ''v'' は[[調和関数|調和]]であることがわかる。

== 関連項目 ==
* [[コーシー・リーマンの関係式]]
* {{仮リンク|コーシー&ndash;リーマン関数|en|CR-function}} (CR-関数)
* {{仮リンク|ドルボー複体|en|Dolbeault complex<!-- リダイレクト先の「[[:en:Dolbeault cohomology]]」は、[[:ja:ドルボーコホモロジー]] とリンク -->}}
* [[ドルボー作用素]]
* {{仮リンク|多重調和関数|en|Pluriharmonic function}}

==脚注==
{{Reflist|30em}}
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| mr = 0265616
| zbl = 0201.10002
}}. "''Areolar derivative and functions of bounded variation''" (free English translation of the title) is an important reference paper in the theory of [[areolar derivative]]s. 
*{{Citation
  | last = Levi
  | first = Eugenio Elia
  | author-link = Eugenio Elia Levi
  | title = Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse
  | language = Italian
  | journal = [[Annali di Matematica Pura e Applicata]]
  | series = s. III
  | volume = XVII
  | issue = 1
  | pages = 61–87
  | year = 1910
  | url = http://www.springerlink.com/content/yr0150m4tq64j465/
  | doi = 10.1007/BF02419336
  | jfm = 41.0487.01
}}. "''Studies on essential singular points of analytic functions of two or more complex variables''" (English translation of the title) is an important paper in the [[Several complex variables|theory of functions of several complex variables]], where the problem of determining what kind of [[hypersurface]] can be the [[Boundary (topology)|boundary]] of a [[domain of holomorphy]].
*{{Citation
 | last = Levi
 | first = Eugenio Elia
 | author-link = Eugenio Elia Levi
 | title = Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse
 | language = Italian
 | journal = [http://www.springerlink.com/content/108198/?p=723d441f3aee4d4bb65e370e90b9c567&pi=0 Annali di Matematica Pura e Applicata]
 | series = s. III,
 | volume = XVIII
 | issue = 1
 | pages = 69–79
 | year = 1911
 | url = http://www.springerlink.com/content/g5734qr179j25m78/
 | doi = 10.1007/BF02420535
 | id = 
 | jfm = 42.0449.02
}}. "''On the hypersurfaces of the 4-dimensional space that can be the boundary of the domain of existence of an analytic function of two complex variables''" (English translation of the title) is another important paper in the [[Several complex variables|theory of functions of several complex variables]], investigating further the theory started in {{harv|Levi|1910}}.
*{{Citation
| last = Levi-Civita
| first = Tullio
| author-link = Tullio Levi Civita
| title = Sulle funzioni di due o più variabili complesse
| language = Italian
| journal = Rendiconti della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| series = 5
| volume = XIV
| issue = 2
| pages = 492–499
| year = 1905
| url = https://archive.org/stream/rendiconti51419052acca#page/492/mode/2up/search/Levi
| jfm = 36.0482.01
}}. "''On the functions of two or more complex variables''" (free English translation of the title) is the first paper where a sufficient condition for the solvability of the [[Cauchy problem]] for [[Several complex variables|holomorphic functions of several complex variables]] is given.
*{{Citation
  | last = Osgood 
  | first = William Fogg
  | author-link = William Fogg Osgood
  | title = Topics in the theory of functions of several complex variables
  | place = New York
  | publisher = [[Dover]]
  | origyear = 1913
  | year = 1966
  | edition = unabridged and corrected
  | pages = IV+120
  | doi = 
  | id = 
  | jfm = 45.0661.02
  | mr = 0201668
  | zbl = 0138.30901
  | isbn = 
}}.
*{{Citation
  | last = Peschl
  | first = Ernst
  | author-link = Ernst Peschl
  | title = Über die Krümmung von Niveaukurven bei der konformen Abbildung einfachzusammenhängender Gebiete auf das Innere eines Kreises. Eine Verallgemeinerung eines Satzes von E. Study.
  | language = German
  | journal = [[Mathematische Annalen]]
  | volume = 106
  | issue =
  | pages = 574–594
  | year = 1932
  | url = http://www.digizeitschriften.de/en/main/dms/img/?PPN=GDZPPN002275570
  | doi = 10.1007/BF01455902
  | id = 
  | jfm = 58.1096.05
  | mr = 1512774
  | zbl = 0004.30001
}}, (in [[German language|German]]) available at [http://www.digizeitschriften.de/ DigiZeitschriften].
*{{Citation
  | last = Poincaré
  | first = H.
  | author-link = Henri Poincaré
  | title = Sur les propriétés du potentiel et sur les fonctions Abéliennes
| language = French
  | journal = [[Acta Mathematica]]
  | volume = 22
  | issue = 1
  | pages = 89–178
  | year = 1899
  | url = http://www.springerlink.com/content/a559gn7k60w08q3w/
  | doi = 10.1007/BF02417872
  | id = 
  | jfm = 29.0370.02
}}.
*{{Citation
| last = Pompeiu
| first = D.
| author-link = Dimitrie Pompeiu
| title = Sur une classe de fonctions d'une variable complexe
| language = French
| journal = [http://www.springerlink.com/content/120943/ Rendiconti del Circolo Matematico di Palermo]
| volume = 33
| issue = 1
| pages = 108–113
| year = 1912
| url = http://www.springerlink.com/content/gm16738uw57245n1/
| doi = 10.1007/BF03015292
| id = 
| jfm = 43.0481.01
}}.
*{{Citation
| last = Pompeiu
| first = D.
| author-link = Dimitrie Pompeiu
| title = Sur une classe de fonctions d'une variable complexe et sur certaines équations intégrales
| language = French
| journal = [http://www.springerlink.com/content/120943/ Rendiconti del Circolo Matematico di Palermo]
| volume =  35
| issue = 1
| pages = 277–281
| year = 1913
| url = http://www.springerlink.com/content/t8m75861w3n30737/
| doi = 10.1007/BF03015607
| id =
}}.
*{{Citation
|last= Vekua
|first= I. N.
|author-link=Ilia Vekua
|title= Generalized Analytic Functions 
|url=
|series= International Series of Monographs in Pure and Applied Mathematics
|volume= 25
|year= 1962
|publisher= [[Pergamon Press]]
|location= London–Paris–Frankfurt
|isbn=
|doi=
|id= 
|mr= 0150320
|zbl= 0100.07603
|pages= xxx+668
}}
*{{Citation
  | last = Wirtinger
  | first = Wilhelm
  | author-link = Wilhelm Wirtinger
  | title = Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen
  | language = German
  | journal = [[Mathematische Annalen]]
  | volume = 97
  | issue =
  | pages = 357–375
  | year = 1926
  | url = http://www.digizeitschriften.de/en/main/dms/img/?PPN=PPN235181684_0097&DMDID=dmdlog19
  | doi = 10.1007/BF01447872
  | id = 
  | jfm = 52.0342.03
}}, available at [http://www.digizeitschriften.de/ DigiZeitschriften]. In this important paper, Wirtinger introduces several important concepts in the [[Several complex variables|theory of functions of several complex variables]], namely [[Wirtinger's derivatives]] and the [[tangential Cauchy-Riemann condition]].
-->
==参考文献==
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  | last = Andreotti
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  | title = Introduzione all'analisi complessa (Lezioni tenute nel febbraio 1972) 
  | language = Italian
  | place = Rome
  | publisher = [[Accademia Nazionale dei Lincei]]
  | year = 1976
  | series = Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni
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  | url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33190
  | doi =
  | id =
  | isbn = }}. ''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|Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni "''Beniamino Segre''"]].
*{{Citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Unification of global and local existence theorems for holomorphic functions of several complex variables
| journal =  Memorie della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| series = 8
| volume = 18
| issue = 3
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}}.
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 | last = Gunning
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 | last2 = Rossi
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 | title = Analytic Functions of Several Complex Variables
 | series = Prentice-Hall series in Modern Analysis
 | place = [[Englewood Cliffs]], N.J.
 | publisher = [[Prentice-Hall]]
 | pages = xiv+317
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}}.
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 | last = Gunning
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 | publisher = Wadsworth & Brooks/Cole
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}}.
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  | last = Hörmander
  | first = Lars
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  | volume = 3
  | pages = XV+349
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}}.
*{{Citation
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  | title = Methods of Complex Analysis in Partial Differential Equations and Applications
  | place = New York–Chichester–Brisbane–Toronto–Singapore
  | publisher = [[John Wiley & Sons]]
  | series = [[Canadian Mathematical Society]] Series of Monographs and Advanced Texts
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  | year = 1988
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}}.
*{{Citation
  | last = Martinelli
  | first = Enzo
  | author-link = 
  | title = Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali 
  | language = Italian
  | place = Rome
  | publisher = [[Accademia Nazionale dei Lincei]]
  | year = 1984
  | series = Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni
  | volume = 67
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  | url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33233
  | doi =
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}}. "''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''".

*{{Citation
| last = Remmert
| first = Reinhold
| author-link = Reinhold Remmert
| title = Theory of Complex Functions
| place = New York–Berlin–Heidelberg–Barcelona–Hong Kong–London–Milan–Paris–Singapore–Tokyo
| publisher = [[Springer Verlag]]
| year = 1991
| series = Graduate Texts in Mathematics
| volume = 122
| edition = Fourth corrected 1998 printing
| pages = xx+453
| url = https://books.google.co.jp/books?id=CC0dQxtYb6kC&printsec=frontcover&redir_esc=y&hl=ja#v=onepage&q&f=true
| doi = 
| id = 
| mr = 1084167
| zbl = 0780.30001
| isbn = 0-387-97195-5
}} ISBN 978-0-387-97195-7. A textbook on [[complex analysis]] including many historical notes on the subject.
*{{Citation
| last = Severi
| first = Francesco
| author-link = 
| title = Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma
| language = Italian
| place = Padova
| publisher = CEDAM – Casa Editrice Dott. Antonio Milani
| year = 1958
| pages = XIV+255
| url =
| doi =
| id =
| zbl= 0094.28002
| isbn = }}. Notes from a course held by Francesco Severi at the [[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''".
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 | first1 = 道夫
 | year = 2003
 | title = 複素関数入門
 | series = 現代数学への入門
 | publisher = [[岩波書店]]
 | isbn = 4-00-006874-1
 | ref = harv
}}
* {{Cite book
 | 和書
 | last1 = 野口
 | first1 = 潤次郎
 | year = 2002
 | title = 複素解析概論
 | series = 数学選書12
 | edition = 第6版
 | publisher = 裳華房
 | isbn = 978-4-7853-1314-2
 | ref = harv
}}

{{DEFAULTSORT:ういるていんかあのひふん}}
[[Category:複素解析]]
[[Category:微分作用素]]
[[Category:解析学]]
[[Category:数学に関する記事]]
[[Category:数学のエポニム]]