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.

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.

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]

Conjugation

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

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