Then he writes the equation defining the functions he calls biharmonique,[5] previously written using
partial derivatives with respect to the
realvariables with ranging from 1 to , exactly in the following way[6]
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.
Definition 1. Consider the
complex plane (in a sense of expressing a complex number for real numbers and ). The Wirtinger derivatives are defined as the following
linearpartial differential operators of first order:
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
functions on a
domain 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 is
complex differentiable at a point, the Wirtinger derivative agrees with the complex derivative . This follows from the
Cauchy-Riemann equations. For the complex function which is complex differentiable
where the third equality uses the Cauchy-Riemann equations .
The second Wirtinger derivative is also related with complex differentiation; is equivalent to the Cauchy-Riemann equations in a complex form.
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 and and two
maps and having natural
smoothness requirements.[17]
^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.
^See (
Poincaré 1899, p. 112), equation 2': note that, throughout the paper, the symbol is used to signify
partial differentiation respect to a given
variable, instead of the now commonplace symbol ∂.
^See problem 2 in
Henrici 1993, p. 294 for one example of such a function.
^See also the excellent book by
Vekua (1962, p. 55), Theorem 1.31: If the generalized derivative , p > 1, then the function has
almost everywhere in a derivative in the sense of
Pompeiu, the latter being equal to the
Generalized derivative in the sense of
Sobolev.
^With or without the attribution of the concept to
Wilhelm Wirtinger: see, for example, the well known monograph
Hörmander 1990, p. 1,23.
^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.
^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.
^In this work, the authors prove some of the properties of Wirtinger derivatives also for the general case of 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.
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,
JFM42.0449.02,
S2CID120133326. "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).
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,
MR0150320,
Zbl0100.07603
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,
MR0917525,
Zbl0705.32006.
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,
Zbl0094.28002. 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".
Then he writes the equation defining the functions he calls biharmonique,[5] previously written using
partial derivatives with respect to the
realvariables with ranging from 1 to , exactly in the following way[6]
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.
Definition 1. Consider the
complex plane (in a sense of expressing a complex number for real numbers and ). The Wirtinger derivatives are defined as the following
linearpartial differential operators of first order:
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
functions on a
domain 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 is
complex differentiable at a point, the Wirtinger derivative agrees with the complex derivative . This follows from the
Cauchy-Riemann equations. For the complex function which is complex differentiable
where the third equality uses the Cauchy-Riemann equations .
The second Wirtinger derivative is also related with complex differentiation; is equivalent to the Cauchy-Riemann equations in a complex form.
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 and and two
maps and having natural
smoothness requirements.[17]
^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.
^See (
Poincaré 1899, p. 112), equation 2': note that, throughout the paper, the symbol is used to signify
partial differentiation respect to a given
variable, instead of the now commonplace symbol ∂.
^See problem 2 in
Henrici 1993, p. 294 for one example of such a function.
^See also the excellent book by
Vekua (1962, p. 55), Theorem 1.31: If the generalized derivative , p > 1, then the function has
almost everywhere in a derivative in the sense of
Pompeiu, the latter being equal to the
Generalized derivative in the sense of
Sobolev.
^With or without the attribution of the concept to
Wilhelm Wirtinger: see, for example, the well known monograph
Hörmander 1990, p. 1,23.
^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.
^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.
^In this work, the authors prove some of the properties of Wirtinger derivatives also for the general case of 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.
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,
JFM42.0449.02,
S2CID120133326. "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).
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,
MR0150320,
Zbl0100.07603
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,
MR0917525,
Zbl0705.32006.
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,
Zbl0094.28002. 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".