In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operatorsSee references and .), 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.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.
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 , as briefly noted by and by .Reference to the work of Henri Poincaré is precisely stated by , while Reinhold Remmert does not cite any reference to support his assertion. In the third paragraph of his 1899 paper,See reference Henri Poincaré first defines the complex variable in and its complex conjugate as follows
Then he writes the equation defining the functions he calls biharmonique,These functions are precisely pluriharmonic functions, and the linear differential operator defining them, i.e. the operator in equation 2 of , is exactly the n-dimensional pluriharmonic operator. previously written using partial derivatives with respect to the real variables with ranging from 1 to , exactly in the following waySee , 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 ∂.
This implies that he implicitly used below: to see this it is sufficient to compare equations 2 and 2' of . Apparently, this paper was not noticed by early researchers in the theory of functions of several complex variables: in the papers of , (and ) and of 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 (first published in 1913),The corrected Dover edition of Osgood's 1913 paper contains much important historical information on the early development of the theory of functions of several complex variables, and is therefore a useful source. 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 operatorSee : curiously, he calls Cauchy–Riemann equations this set of equations. 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 , a new step in the definition of the concept was taken by Dimitrie Pompeiu: in the paper , given a complex valued differentiable function (in the sense of real analysis) of one complex variable defined in the neighbourhood of a given point he defines the areolar derivative as the following limit
where is the boundary of a disk of radius entirely contained in the domain of definition of i.e. his bounding circle.This is the definition given by in his approach to Pompeiu's work: as remarks, the original definition of does not require the domain of integration to be a circle. See the entry areolar derivative for further information. This is evidently an alternative definition of Wirtinger derivative respect to the complex conjugate variable:See the section "Formal definition" of this entry. it is a more general one, since, as noted a by , the limit may exist for functions that are not even differentiable at See problem 2 in for one example of such a function. According to , the first to identify the areolar derivative as a weak derivative in the sense of Sobolev was Ilia Vekua.See also the excellent book by , 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 . In his following paper, 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 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,With or without the attribution of the concept to Wilhelm Wirtinger: see, for example, the well known monograph . 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 ,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. the monograph of ,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. and the monograph of 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. which are used as general references in this and the following sections.
Functions of one complex variable Consider the complex plane (in a sense of expressing a complex number for real numbers and ). Let represent the complex conjugate of Then the Wirtinger derivatives are defined as the following linear partial differential operators of first order:
Clearly, the natural domain of definition of these partial differential operators is the space of functions on a domain but, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions
The operators can be derived by noting that , and taking the derivative with respect to using the multivariable chain rule, treating as intermediate variables.
Functions of n > 1 complex variables Consider the Euclidean space on the complex field The Wirtinger derivatives are defined as the following linear partial 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 first definition of Wirtinger's derivatives for and .
It can also be done through actual application of the Cauchy-Riemann equations.
The final equality comes from it being one of four equivalent formulations of the complex derivative through partial derivatives of the components.
The second Wirtinger derivative is also related with complex differentiation; 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 is a complex vector and that where are real vectors, with n ≥ 1: also it is assumed that the subset can be thought of as a domain in the real euclidean space or in its isomorphic complex counterpart All the proofs are easy consequences of and and of the corresponding properties of the derivatives (ordinary or partial).
Linearity If and are complex numbers, then for the following equalities hold
Product rule If then for the product rule holds
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 and and two maps and having natural smoothness requirements.See and also : Gunning considers the general case of functions but only for p = 1. References and , as already pointed out, consider only holomorphic maps with p = 1: however, the resulting formulas are formally very similar.
Functions of one complex variable If and then the chain rule holds
Functions of n > 1 complex variables If and then for the following form of the chain rule holds
Conjugation If then for the following equalities hold
See also CR–function Dolbeault complex Dolbeault operator Pluriharmonic function
Notes
References
Historical references . "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. . . "Areolar derivative and functions of bounded variation" (free English translation of the title) is an important reference paper in the theory of areolar derivatives. . "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. . "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 . . "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. . , available at DigiZeitschriften. . . .
, 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 . 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". . . . . . . .