In
complex analysis of one and
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
, Wirtinger derivatives (sometimes also called Wirtinger operators), 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 operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s of the first order which behave in a very similar manner to the ordinary
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s with respect to one
real variable, when applied to
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s,
antiholomorphic functions or simply
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s 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.
Historical notes
Early days (1899–1911): the work of Henri Poincaré
Wirtinger derivatives were used in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
at least as early as in the paper , as briefly noted by and by . As a matter of fact, in the third paragraph of his 1899 paper,
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
first defines the
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
in
and its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
as follows
:
Then he writes the equation defining the functions
he calls ''biharmonique'', previously written using
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s with respect to the
real variables with
ranging from 1 to
, exactly in the following way
:
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 operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s of the theory are expressed directly by using
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
respect to the
real and
imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s of the
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
s involved. In the long survey paper by (first published in 1913),
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
with respect to each
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
of a
holomorphic function of several complex variables seem to be meant as
formal derivative
In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal deriva ...
s: as a matter of fact when
Osgood expresses the
pluriharmonic operator and the
Levi operator, he follows the established practice of
Amoroso,
Levi
Levi (; ) was, according to the Book of Genesis, the third of the six sons of Jacob and Leah (Jacob's third son), and the founder of the Israelite Tribe of Levi (the Levites, including the Kohanim) and the great-grandfather of Aaron, Moses and ...
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 mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
(in the sense of
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include con ...
) of one
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
defined in the
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of a given
point he defines the
areolar derivative as the following
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
:
where
is the
boundary of a
disk of radius
entirely contained in the
domain of definition of
i.e. his bounding
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
. This is evidently an alternative definition of Wirtinger derivative respect to the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
variable: it is a more general one, since, as noted a by , the limit may exist for functions that are not even
differentiable at
According to , the first to identify the
areolar derivative as a
weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b.
The method ...
in the
sense of Sobolev was
Ilia Vekua
Ilia Vekua ( ka, ილია ვეკუა; russian: link=no, Илья́ Не́сторович Ве́куа; 23 April 1907 in the village of Shesheleti, Kutais Governorate, Russian Empire (modern day Gali District, Abkhazia, Republic of Georg ...
. In his following paper, uses this newly defined concept in order to introduce his generalization of
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
, 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, 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 , the
monograph of , and the monograph of which are used as general references in this and the following sections.
Functions of one complex variable
Consider the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
The Wirtinger derivatives are defined as the following
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
partial differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s of first order:
:
Clearly, the natural
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
of definition of these partial differential operators is the space of
functions on a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
but, since these operators are
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and have
constant coefficients, they can be readily extended to every
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
of
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functio ...
s.
Functions of ''n'' > 1 complex variables
Consider the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
on the
complex field The Wirtinger derivatives are defined as the following
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
partial differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s of first order:
As for Wirtinger derivatives for functions of one complex variable, the natural
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
of definition of these partial differential operators is again the space of
functions on a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
and again, since these operators are
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and have
constant coefficients, they can be readily extended to every
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
of
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functio ...
s.
Basic properties
In the present section and in the following ones it is assumed that
is a
complex vector and that
where
are
real vector
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s, with ''n'' ≥ 1: also it is assumed that the
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
can be thought of as a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
in the
real euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
or in its
isomorphic complex counterpart
All the proofs are easy consequences of and and of the corresponding properties of the
derivatives
The derivative of a function is the rate of change of the function's output relative to its input value.
Derivative may also refer to:
In mathematics and economics
*Brzozowski derivative in the theory of formal languages
*Formal derivative, an ...
(ordinary or
partial).
Linearity
If
and
are
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, then for
the following equalities hold
:
Product rule
If
then for
the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
holds
:
This property implies that Wirtinger derivatives are
derivations from the
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
point of view, exactly like ordinary
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s are.
Chain rule
This property takes two different forms respectively for functions of one and
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
: for the ''n'' > 1 case, to express the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
in its full generality it is necessary to consider two
domains and
and two
maps
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Althoug ...
and
having natural
smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
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
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
holds
:
Functions of ''n'' > 1 complex variables
If
and
then for
the following form of the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
holds
:
Conjugation
If
then for
the following equalities hold
:
See also
*
CR–function
*
Dolbeault complex In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohomo ...
*
Dolbeault operator
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex number, complex coefficients.
Complex forms have broad applications in differential geometry. On comp ...
*
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
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Eucl ...
can be the
boundary of a
domain of holomorphy
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain.
Formal ...
.
*. "''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 a
DigiZeitschriften
*.
*.
*.
*
*, available a
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''".
*.
*.
*.
*.
*.
*.
*.
*. "''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''".
* . A textbook on
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
including many historical notes on the subject.
*. 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''".
{{DEFAULTSORT:Wirtinger Derivatives
Complex analysis
Differential operators
Mathematical analysis