In
complex analysis of one and
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...
, 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 operators of the first order which behave in a very similar manner to the ordinary
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
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 algebraic ...
at least as early as in the paper , as briefly noted by and by . In the third paragraph of his 1899 paper,
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
first defines the
complex variable 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 numbers, then the complex conjugate of a + bi is 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). P ...
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 operators of the theory are expressed directly by using
partial derivatives 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 form ...
s of the
complex variables involved. In the long survey paper by (first published in 1913),
partial derivatives with respect to each
complex variable 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 deriv ...
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 Israelites, Israelite Tribe of Levi (the Levites, including the Kohanim) and the great-grandfather of Aaron ...
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 co ...
) of one
complex variable defined in the
neighbourhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
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
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain of ...
of
i.e. his bounding
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
. 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 numbers, then the complex conjugate of a + bi is a - ...
variable: it is a more general one, since, as noted a by , the limit may exist for functions that are not even
differentiable
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 ...
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 o ...
in the
sense of Sobolev was
Ilia Vekua. 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 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 retur ...
s, 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
A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...
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 (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
(in a sense of expressing a complex number
for real numbers
and
). The Wirtinger derivatives are defined as the following
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
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
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
and have
constant coefficients
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form
a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x)
where and are arbi ...
, they can be readily extended to every
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
of
generalized functions.
Functions of ''n'' > 1 complex variables
Consider the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
on the
complex field The Wirtinger derivatives are defined as the following
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
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
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
and have
constant coefficients
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form
a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x)
where and are arbi ...
, they can be readily extended to every
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
of
generalized functions.
Relation with complex differentiation
When a function
is
complex differentiable
In mathematics, a holomorphic function is a complex-valued function of one or Function of several complex variables, more complex number, complex variables that is Differentiable function#Differentiability in complex analysis, complex differ ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and ...
and that
where
are
real vectors, with ''n'' ≥ 1: also it is assumed that the
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
can be thought of as a
domain in the
real euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
or in its
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
counterpart
All the proofs are easy consequences of and and of the corresponding properties of the
derivatives (ordinary or
partial
Partial may refer to:
Mathematics
*Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...
).
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, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
point of view, exactly like ordinary
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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 functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...
: 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 ...
in its full generality it is necessary to consider two
domains and
and two
maps
A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
and
having natural
smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
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 ...
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 ...
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
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 Euclidea ...
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.
Forma ...
.
*. "''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
The (; literally the "Academy of the Lynx-Eyed"), anglicised as the Lincean Academy, is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy. Founded in ...
, 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 algebraic ...
including many historical notes on the subject.
*. Notes from a course held by Francesco Severi at the
Istituto Nazionale di Alta Matematica
The Istituto Nazionale di Alta Matematica Francesco Severi, abbreviated as INdAM, is a government created non-profit research institution whose main purpose is to promote research in the field of mathematics and its applications and the diffusion ...
(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