Wirtinger derivatives

In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators ^[1] ), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables. ^[2]

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 ( Poincaré 1899 ), as briefly noted by Cherry & Ye (2001 , p. 31) and by Remmert (1991 , pp. 66–67). ^[3] In the third paragraph of his 1899 paper, ^[4] Henri Poincaré first defines the complex variable in ${\displaystyle \mathbb {C} ^{n}}$ and its complex conjugate as follows

${\displaystyle {\begin{cases}x_{k}+iy_{k}=z_{k}\\x_{k}-iy_{k}=u_{k}\end{cases}}\qquad 1\leqslant k\leqslant n.}$

Then he writes the equation defining the functions ${\displaystyle V}$ he calls biharmonique, ^[5] previously written using partial derivatives with respect to the real variables ${\displaystyle x_{k},y_{q}}$ with ${\displaystyle k,q}$ ranging from 1 to ${\displaystyle n}$, exactly in the following way ^[6]

${\displaystyle {\frac {d^{2}V}{dz_{k}\,du_{q}}}=0}$

This implies that he implicitly used definition 2 below: to see this it is sufficient to compare equations 2 and 2' of ( Poincaré 1899 , p. 112). Apparently, this paper was not noticed by early researchers in the theory of functions of several complex variables: in the papers of Levi-Civita (1905), Levi (1910) (and Levi 1911) and of Amoroso (1912) all fundamental partial differential operators of the theory are expressed directly by using partial derivatives respect to the real and imaginary parts of the complex variables involved. In the long survey paper by Osgood (1966) (first published in 1913), ^[7] partial derivatives with respect to each complex variable of a holomorphic function of several complex variables seem to be meant as formal derivatives: as a matter of fact when Osgood expresses the pluriharmonic operator ^[8] and the Levi operator, he follows the established practice of Amoroso, Levi and Levi-Civita.

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

According to Henrici (1993 , p. 294), a new step in the definition of the concept was taken by Dimitrie Pompeiu: in the paper ( Pompeiu 1912 ), given a complex valued differentiable function (in the sense of real analysis) of one complex variable ${\displaystyle g(z)}$ defined in the neighbourhood of a given point ${\displaystyle z_{0}\in \mathbb {C} ,}$ he defines the areolar derivative as the following limit

${\displaystyle {{\frac {\partial g}{\partial {\bar {z}}}}(z_{0})}\mathrel {\overset {\mathrm {def} }{=}} \lim _{r\to 0}{\frac {1}{2\pi ir^{2}}}\oint _{\Gamma (z_{0},r)}g(z)\mathrm {d} z,}$

where ${\displaystyle \Gamma (z_{0},r)=\partial D(z_{0},r)}$ is the boundary of a disk of radius ${\displaystyle r}$ entirely contained in the domain of definition of ${\displaystyle g(z),}$ i.e. his bounding circle. ^[9] This is evidently an alternative definition of Wirtinger derivative respect to the complex conjugate variable: ^[10] it is a more general one, since, as noted a by Henrici (1993 , p. 294), the limit may exist for functions that are not even differentiable at ${\displaystyle z=z_{0}.}$ ^[11] According to Fichera (1969 , p. 28), the first to identify the areolar derivative as a weak derivative in the sense of Sobolev was Ilia Vekua. ^[12] In his following paper, 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 Wirtinger 1927 in order to simplify the calculations of quantities occurring in the theory of functions of several complex variables: as a result of the introduction of these differential operators, the form of all the differential operators commonly used in the theory, like the Levi operator and the 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.

Formal definition

Despite their ubiquitous use, ^[13] it seems that there is no text listing all the properties of Wirtinger derivatives: however, fairly complete references are the short course on multidimensional complex analysis by Andreotti (1976 , pp. 3–5), ^[14] the monograph of Gunning & Rossi (1965 , pp. 3–6), ^[15] and the monograph of Kaup & Kaup (1983 , p. 2,4) ^[16] which are used as general references in this and the following sections.

Functions of one complex variable

Definition 1. Consider the complex plane ${\displaystyle \mathbb {C} \equiv \mathbb {R} ^{2}=\{(x,y)\mid x,y\in \mathbb {R} \}}$ (in a sense of expressing a complex number ${\displaystyle z=x+iy}$ for real numbers ${\displaystyle x}$ and ${\displaystyle y}$). The Wirtinger derivatives are defined as the following linear partial differential operators of first order:

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial z}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\\{\frac {\partial }{\partial {\bar {z}}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)\end{aligned}}}$

Clearly, the natural domain of definition of these partial differential operators is the space of ${\displaystyle C^{1}}$ functions on a domain ${\displaystyle \Omega \subseteq \mathbb {R} ^{2},}$ but, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions.

Functions of n > 1 complex variables

Definition 2. Consider the Euclidean space on the complex field

$${\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{2n}=\left\{\left(\mathbf {x} ,\mathbf {y} \right)=\left(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\right)\mid \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}\right\}.}$$

The Wirtinger derivatives are defined as the following linear partial differential operators of first order:

$${\displaystyle {\begin{cases}{\frac {\partial }{\partial z_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}-i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial z_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}-i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}},\qquad {\begin{cases}{\frac {\partial }{\partial {\bar {z}}_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}+i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial {\bar {z}}_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}+i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}}.}$$

As for Wirtinger derivatives for functions of one complex variable, the natural domain of definition of these partial differential operators is again the space of ${\displaystyle C^{1}}$ functions on a domain ${\displaystyle \Omega \subset \mathbb {R} ^{2n},}$ and again, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions.

Relation with complex differentiation

When a function ${\displaystyle f}$ is complex differentiable at a point, the Wirtinger derivative ${\displaystyle \partial f/\partial z}$ agrees with the complex derivative ${\displaystyle df/dz}$. This follows from the Cauchy-Riemann equations. For the complex function ${\displaystyle f(z)=u(z)+iv(z)}$ which is complex differentiable

${\displaystyle {\begin{aligned}{\frac {\partial f}{\partial z}}&={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-i{\frac {\partial f}{\partial y}}\right)\\&={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}-i{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial y}}\right)\\&={\frac {\partial u}{\partial z}}+i{\frac {\partial v}{\partial z}}={\frac {df}{dz}}\end{aligned}}}$

where the third equality uses the Cauchy-Riemann equations ${\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}}$.

The second Wirtinger derivative is also related with complex differentiation; ${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}$ is equivalent to the Cauchy-Riemann equations in a complex form.

Basic properties

In the present section and in the following ones it is assumed that ${\displaystyle z\in \mathbb {C} ^{n}}$ is a complex vector and that ${\displaystyle z\equiv (x,y)=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})}$ where ${\displaystyle x,y}$ are real vectors, with n ≥ 1: also it is assumed that the subset ${\displaystyle \Omega }$ can be thought of as a domain in the real euclidean space ${\displaystyle \mathbb {R} ^{2n}}$ or in its isomorphic complex counterpart ${\displaystyle \mathbb {C} ^{n}.}$ All the proofs are easy consequences of definition 1 and definition 2 and of the corresponding properties of the derivatives (ordinary or partial).

Linearity

Lemma 1. If ${\displaystyle f,g\in C^{1}(\Omega )}$ and ${\displaystyle \alpha ,\beta }$ are complex numbers, then for ${\displaystyle i=1,\dots ,n}$ the following equalities hold

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial z_{i}}}+\beta {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial {\bar {z}}_{i}}}+\beta {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}}$

Product rule

Lemma 2. If ${\displaystyle f,g\in C^{1}(\Omega ),}$ then for ${\displaystyle i=1,\dots ,n}$ the product rule holds

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}(f\cdot g)&={\frac {\partial f}{\partial z_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}(f\cdot g)&={\frac {\partial f}{\partial {\bar {z}}_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}}$

This property implies that Wirtinger derivatives are derivations from the abstract algebra point of view, exactly like ordinary derivatives are.

Chain rule

This property takes two different forms respectively for functions of one and several complex variables: for the n > 1 case, to express the chain rule in its full generality it is necessary to consider two domains ${\displaystyle \Omega '\subseteq \mathbb {C} ^{m}}$ and ${\displaystyle \Omega ''\subseteq \mathbb {C} ^{p}}$ and two maps ${\displaystyle g:\Omega '\to \Omega }$ and ${\displaystyle f:\Omega \to \Omega ''}$ having natural smoothness requirements. ^[17]

Functions of one complex variable

Lemma 3.1 If ${\displaystyle f,g\in C^{1}(\Omega ),}$ and ${\displaystyle g(\Omega )\subseteq \Omega ,}$ then the chain rule holds

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial z}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial z}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial z}}\\{\frac {\partial }{\partial {\bar {z}}}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial {\bar {z}}}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial {\bar {z}}}}\end{aligned}}}$

Functions of n > 1 complex variables

Lemma 3.2 If ${\displaystyle g\in C^{1}(\Omega ',\Omega )}$ and ${\displaystyle f\in C^{1}(\Omega ,\Omega ''),}$ then for ${\displaystyle i=1,\dots ,m}$ the following form of the chain rule holds

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial z_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial {\bar {z}}_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial {\bar {z}}_{i}}}\end{aligned}}}$

Conjugation

Lemma 4. If ${\displaystyle f\in C^{1}(\Omega ),}$ then for ${\displaystyle i=1,\dots ,n}$ the following equalities hold

${\displaystyle {\begin{aligned}{\overline {\left({\frac {\partial f}{\partial z_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial {\bar {z}}_{i}}}\\{\overline {\left({\frac {\partial f}{\partial {\bar {z}}_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial z_{i}}}\end{aligned}}}$

See also

• CR–function
• Dolbeault complex
• Dolbeault operator
• Pluriharmonic function

Notes

1. See references Fichera 1986 , p. 62 and Kracht & Kreyszig 1988 , p. 10.
2. Some of the basic properties of Wirtinger derivatives are the same ones as the properties characterizing the ordinary (or partial) derivatives and used for the construction of the usual differential calculus.
3. Reference to the work Poincaré 1899 of Henri Poincaré is precisely stated by Cherry & Ye (2001), while Reinhold Remmert does not cite any reference to support his assertion.
4. See reference ( Poincaré 1899 , pp. 111–114)
5. These functions are precisely pluriharmonic functions, and the linear differential operator defining them, i.e. the operator in equation 2 of ( Poincaré 1899 , p. 112), is exactly the n-dimensional pluriharmonic operator.
6. See ( Poincaré 1899 , p. 112), equation 2': note that, throughout the paper, the symbol ${\displaystyle d}$ is used to signify partial differentiation respect to a given variable, instead of the now commonplace symbol ∂.
7. The corrected Dover edition of the paper ( Osgood 1913 ) harv error: no target: CITEREFOsgood1913 (help) contains much important historical information on the early development of the theory of functions of several complex variables, and is therefore a useful source.
8. See Osgood (1966 , pp. 23–24): curiously, he calls Cauchy–Riemann equations this set of equations.
9. This is the definition given by Henrici (1993 , p. 294) in his approach to Pompeiu's work: as Fichera (1969 , p. 27) remarks, the original definition of Pompeiu (1912) does not require the domain of integration to be a circle. See the entry areolar derivative for further information.
10. See the section "Formal definition" of this entry.
11. See problem 2 in Henrici 1993 , p. 294 for one example of such a function.
12. See also the excellent book by Vekua (1962 , p. 55), Theorem 1.31: If the generalized derivative ${\displaystyle \partial _{\bar {z}}w\in }$ ${\displaystyle L_{p}(\Omega )}$, p > 1, then the function ${\displaystyle w(z)}$ has almost everywhere in ${\displaystyle G}$ a derivative in the sense of Pompeiu, the latter being equal to the Generalized derivative in the sense of Sobolev ${\displaystyle \partial _{\bar {z}}w}$.
13. With or without the attribution of the concept to Wilhelm Wirtinger: see, for example, the well known monograph Hörmander 1990 , p. 1,23.
14. In this course lectures, Aldo Andreotti uses the properties of Wirtinger derivatives in order to prove the closure of the algebra of holomorphic functions under certain operations: this purpose is common to all references cited in this section.
15. This is a classical work on the theory of functions of several complex variables dealing mainly with its 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.
16. In this work, the authors prove some of the properties of Wirtinger derivatives also for the general case of ${\displaystyle C^{1}}$ 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.
17. See Kaup & Kaup 1983 , p. 4 and also Gunning 1990 , p. 5: Gunning considers the general case of ${\displaystyle C^{1}}$ functions but only for p = 1. References Andreotti 1976 , p. 5 and Gunning & Rossi 1965 , p. 6, as already pointed out, consider only holomorphic maps with p = 1: however, the resulting formulas are formally very similar.

References

Historical references

• Amoroso, Luigi (1912), "Sopra un problema al contorno", Rendiconti del Circolo Matematico di Palermo (in Italian), 33 (1): 75–85, doi:10.1007/BF03015289, JFM   43.0453.03, S2CID   122956910 . "On a boundary value problem" (free translation of the title) is the first paper where a set of (fairly complicate) necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables is given.
• Cherry, W.; Ye, Z. (2001), Nevanlinna's theory of value distribution: the second main theorem and its error terms, Springer Monographs in Mathematics, Berlin: Springer Verlag, pp. XII+202, ISBN   978-3-540-66416-1, MR   1831783, Zbl   0981.30001 .
• Fichera, Gaetano (1969), "Derivata areolare e funzioni a variazione limitata", Revue Roumaine de Mathématiques Pures et Appliquées (in Italian), XIV (1): 27–37, 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 derivatives.
• Levi, Eugenio Elia (1910), "Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse", Annali di Matematica Pura ed Applicata , s. III (in Italian), XVII (1): 61–87, doi:10.1007/BF02419336, JFM   41.0487.01, S2CID   122678686 . "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 theory of functions of several complex variables, where the problem of determining what kind of hypersurface can be the boundary of a domain of holomorphy.
• Levi, Eugenio Elia (1911), "Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse", Annali di Matematica Pura ed Applicata , s. III (in Italian), XVIII (1): 69–79, doi:10.1007/BF02420535, JFM   42.0449.02, S2CID   120133326 . "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 theory of functions of several complex variables, investigating further the theory started in (Levi 1910).
• Levi-Civita, Tullio (1905), "Sulle funzioni di due o più variabili complesse", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 5 (in Italian), XIV (2): 492–499, 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 holomorphic functions of several complex variables is given.
• Osgood, William Fogg (1966) [1913], Topics in the theory of functions of several complex variables (unabridged and corrected ed.), New York: Dover, pp. IV+120, JFM   45.0661.02, MR   0201668, Zbl   0138.30901 .
• Peschl, Ernst (1932), "Über die Krümmung von Niveaukurven bei der konformen Abbildung einfachzusammenhängender Gebiete auf das Innere eines Kreises. Eine Verallgemeinerung eines Satzes von E. Study.", Mathematische Annalen (in German), 106: 574–594, doi:10.1007/BF01455902, JFM   58.1096.05, MR   1512774, S2CID   127138808, Zbl   0004.30001 , available at DigiZeitschriften.
• Poincaré, H. (1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes", Acta Mathematica (in French), 22 (1): 89–178, doi: 10.1007/BF02417872 , JFM   29.0370.02 .
• Pompeiu, D. (1912), "Sur une classe de fonctions d'une variable complexe", Rendiconti del Circolo Matematico di Palermo (in French), 33 (1): 108–113, doi:10.1007/BF03015292, JFM   43.0481.01, S2CID   120717465 .
• Pompeiu, D. (1913), "Sur une classe de fonctions d'une variable complexe et sur certaines équations intégrales", Rendiconti del Circolo Matematico di Palermo (in French), 35 (1): 277–281, doi:10.1007/BF03015607, S2CID   121616964 .
• Vekua, I. N. (1962), Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, vol. 25, London–Paris–Frankfurt: Pergamon Press, pp. xxx+668, MR   0150320, Zbl   0100.07603
• Wirtinger, Wilhelm (1927), "Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen", Mathematische Annalen (in German), 97: 357–375, doi:10.1007/BF01447872, JFM   52.0342.03, S2CID   121149132 , available at DigiZeitschriften. In this important paper, Wirtinger introduces several important concepts in the theory of functions of several complex variables, namely Wirtinger's derivatives and the tangential Cauchy-Riemann condition.

Scientific references

• Andreotti, Aldo (1976), Introduzione all'analisi complessa (Lezioni tenute nel febbraio 1972), Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 24, Rome: Accademia Nazionale dei Lincei, p. 34, archived from the original on 2012-03-07, retrieved 2010-08-28. Introduction to complex analysis is a short course in the theory of functions of several complex variables, held in 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, Prentice-Hall series in Modern Analysis, Englewood Cliffs, N.J.: Prentice-Hall, pp. xiv+317, ISBN   9780821869536, MR   0180696, Zbl   0141.08601 .
• 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, Wiley Classics Library (Reprint ed.), New York–Chichester–Brisbane–Toronto–Singapore: John Wiley & Sons, pp. X+637, ISBN   0-471-58986-1, MR   0822470, Zbl   1107.30300 .
• Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 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, de Gruyter Studies in Mathematics, vol. 3, Berlin–New York: Walter de Gruyter, pp. XV+349, ISBN   978-3-11-004150-7, MR   0716497, Zbl   0528.32001 .
• 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), Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali, Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 67, Rome: Accademia Nazionale dei Lincei, pp. 236+II, archived from the original on 2011-09-27, retrieved 2010-08-24. "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, Graduate Texts in Mathematics, vol. 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 ISBN   978-0-387-97195-7. A textbook on complex analysis including many historical notes on the subject.
• Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255, Zbl   0094.28002 . 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".