Wirtinger Derivative
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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 \Complex^n 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 :\begin x_k+iy_k=z_k\\ x_k-iy_k=u_k \end \qquad 1 \leqslant k \leqslant n. Then he writes the equation defining the functions V 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 x_k, y_q with k, q ranging from 1 to n, exactly in the following way :\frac=0 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 g(z) 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 z_0 \in \Complex, he defines the areolar derivative as the following limit :\mathrel\lim_\frac \oint_ g(z)\mathrmz, where \Gamma(z_0,r)=\partial D(z_0,r) is the boundary of a disk of radius r 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 g(z), 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 z=z_0. 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 ...
\Complex \equiv \R^2 = \ (in a sense of expressing a complex number z = x + iy for real numbers x and y). 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: :\begin \frac &= \frac \left( \frac - i \frac \right) \\ \frac &= \frac \left( \frac + i \frac \right) \end Clearly, the natural domain of definition of these partial differential operators is the space of C^1 functions on a domain \Omega \subseteq \R^2, 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 \Complex^n = \R^ = \left\. 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: \begin \frac = \frac \left( \frac- i \frac \right) \\ \qquad \vdots \\ \frac = \frac \left( \frac- i \frac \right) \\ \end, \qquad \begin \frac = \frac \left( \frac+ i \frac \right) \\ \qquad \vdots \\ \frac = \frac \left( \frac+ i \frac \right) \\ \end. 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 C^1 functions on a domain \Omega \subset \R^, 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 f 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 \partial f/\partial z agrees with the complex derivative df/dz. This follows from the Cauchy-Riemann equations. For the complex function f(z) = u(z) + iv(z) which is complex differentiable :\begin \frac &= \frac \left( \frac - i \frac \right) \\ &= \frac \left( \frac + i \frac -i \frac + \frac \right) \\ &= \frac + i \frac = \frac \end where the third equality uses the first definition of Wirtinger's derivatives for u and v. It can also be done through actual application of the Cauchy-Riemann equations. :\begin \frac &= \frac \left( \frac - i \frac \right) \\ &= \frac \left( \frac + i \frac -i \frac + \frac \right) \\ &= \frac \left( \frac + i \frac + i \frac + \frac \right) \\ &= \frac + i \frac = \frac \end 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; \frac = 0 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 z \in \Complex^n 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 z \equiv (x,y) = (x_1,\ldots,x_n,y_1,\ldots,y_n) where x,y 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 ...
\Omega 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 ...
\R^ 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 \Complex^n. 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 f,g \in C^1(\Omega) and \alpha,\beta 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 i=1,\dots,n the following equalities hold :\begin \frac \left(\alpha f+\beta g\right) &= \alpha\frac + \beta\frac \\ \frac \left(\alpha f+\beta g\right) &= \alpha\frac + \beta\frac \end


Product rule

If f,g \in C^1(\Omega), then for i= 1,\dots,n 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 :\begin \frac (f\cdot g) &= \frac\cdot g + f\cdot\frac \\ \frac (f\cdot g) &= \frac\cdot g + f\cdot\frac \end 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 \Omega'\subseteq\Complex^m and \Omega''\subseteq\Complex^p 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 ...
g: \Omega'\to\Omega and f:\Omega \to \Omega'' 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 C^1 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 f,g \in C^1(\Omega), and g(\Omega ) \subseteq \Omega, 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 :\begin \frac (f\circ g) &= \left(\frac\circ g \right) \frac + \left(\frac\circ g \right) \frac \\ \frac (f\circ g) &= \left(\frac\circ g \right)\frac+ \left(\frac\circ g \right) \frac \end


Functions of ''n'' > 1 complex variables

If g \in C^1(\Omega',\Omega) and f \in C^1(\Omega,\Omega''), then for i= 1,\dots,n 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 :\begin \frac \left(f\circ g\right) &= \sum_^n\left(\frac\circ g \right) \frac + \sum_^n\left(\frac\circ g \right) \frac \\ \frac \left(f\circ g\right) &= \sum_^n\left(\frac\circ g \right) \frac + \sum_^n\left(\frac\circ g \right)\frac \end


Conjugation

If f\in C^1(\Omega), then for i=1,\dots,n the following equalities hold :\begin \overline &= \frac \\ \overline &= \frac \end


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