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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a holomorphic function is a
complex-valued function 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 ...
of one or
more More or Mores may refer to: Computing * MORE (application), outline software for Mac OS * more (command), a shell command * MORE protocol, a routing protocol * Missouri Research and Education Network Music Albums * ''More!'' (album), by Booka S ...
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 ...
variables that is
complex differentiable 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 derivati ...
in a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of each point in 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
complex coordinate space In mathematics, the ''n''-dimensional complex coordinate space (or complex ''n''-space) is the set of all ordered ''n''-tuples of complex numbers. It is denoted \Complex^n, and is the ''n''-fold Cartesian product of the complex plane \Complex wi ...
. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is
infinitely differentiable 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 it ...
and locally equal to its own
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
(''analytic''). Holomorphic functions are the central objects of study in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. Though the term ''
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its
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 ...
. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regular functions''. A holomorphic function whose domain is the whole
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 the ...
is called an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
. The phrase "holomorphic at a point " means not just differentiable at , but differentiable everywhere within some neighbourhood of in the complex plane.


Definition

Given a complex-valued function of a single complex variable, the derivative of at a point in its domain is defined as the
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 ...
:f'(z_0) = \lim_ . This is the same definition as for the
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. F ...
of a
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
, except that all quantities are complex. In particular, the limit is taken as the complex number tends to , and this means that the same value is obtained for any sequence of complex values for that tends to . If the limit exists, is said complex differentiable at . This concept of complex differentiability shares several properties with real differentiability: it is
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 r ...
and obeys 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 + ...
,
quotient rule In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the derivat ...
, and
chain rule In calculus, the chain rule is a formula that expresses the derivative of the 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)=f(g(x)) for every , ...
. A function is holomorphic on an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
if it is ''complex differentiable'' at ''every'' point of . A function is ''holomorphic'' at a point if it is holomorphic on some
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of . A function is ''holomorphic'' on some non-open set if it is holomorphic at every point of . A function may be complex differentiable at a point but not holomorphic at this point. For example, the function is not complex differentiable at , but is complex differentiable elsewhere. So, it is ''not'' holomorphic at . The relationship between real differentiability and complex differentiability is the following: If a complex function is holomorphic, then and have first partial derivatives with respect to and , and satisfy the
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differ ...
:Markushevich, A.I.,''Theory of Functions of a Complex Variable'' (Prentice-Hall, 1965). hree volumes./ref> :\frac = \frac \qquad \mbox \qquad \frac = -\frac\, or, equivalently, the
Wirtinger derivative In complex analysis of one and several complex variables, 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 sev ...
of with 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 - ...
of , is zero: :\frac = 0, which is to say that, roughly, is functionally independent from , the complex conjugate of . If continuity is not given, the converse is not necessarily true. A simple converse is that if and have ''continuous'' first partial derivatives and satisfy the Cauchy–Riemann equations, then is holomorphic. A more satisfying converse, which is much harder to prove, is the
Looman–Menchoff theorem In the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations. I ...
: if is continuous, and have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then is holomorphic.


Terminology

The term ''holomorphic'' was introduced in 1875 by Charles Briot and
Jean-Claude Bouquet Jean-Claude Bouquet (7 September 1819 – 9 September 1885) was a French mathematician who worked with Charles Briot on doubly periodic functions. Bouquet became friends with Briot at the Lycée and wanted to become a mathematics teacher. Biogr ...
, two of
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
's students, and derives from the Greek ὅλος (''hólos'') meaning "whole", and μορφή (''morphḗ'') meaning "form" or "appearance" or "type", in contrast to the term ''
meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
'' derived from μέρος (''méros'') meaning "part". A holomorphic function resembles an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
("whole") in 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 ...
of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated
poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
), resembles a rational fraction ("part") of entire functions in a domain of the complex plane. Cauchy had instead used the term ''synectic''. Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.


Properties

Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero. That is, if functions and are holomorphic in a domain , then so are , , , and . Furthermore, is holomorphic if has no zeros in , or is
meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
otherwise. If one identifies with the real
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differ ...
, a set of two
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s. Every holomorphic function can be separated into its real and imaginary parts , and each of these is a
harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...
on (each satisfies
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
), with the
harmonic conjugate In mathematics, a real-valued function u(x,y) defined on a connected open set \Omega \subset \R^2 is said to have a conjugate (function) v(x,y) if and only if they are respectively the real and imaginary parts of a holomorphic function f(z) of ...
of . Conversely, every harmonic function on a
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
domain is the real part of a holomorphic function: If is the harmonic conjugate of , unique up to a constant, then is holomorphic.
Cauchy's integral theorem In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
implies that the
contour integral In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
of every holomorphic function along a
loop Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, an ...
vanishes: :\oint_\gamma f(z)\,dz = 0. Here is a
rectifiable path Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...
in a simply connected complex domain whose start point is equal to its end point, and is a holomorphic function.
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 ...
states that every function holomorphic inside a
disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...
is completely determined by its values on the disk's boundary. Furthermore: Suppose is a complex domain, is a holomorphic function and the closed disk is completely contained in . Let be the circle forming the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of . Then for every in the interior of : :f(a) = \frac \oint_\gamma \frac\,dz where the contour integral is taken
counter-clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
. The derivative can be written as a contour integral using
Cauchy's differentiation 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 ...
: :f'(a) = \oint_\gamma \,dz, for any simple loop positively winding once around , and :f'(a) = \lim\limits_\frac i\oint_f(z)\,d\bar z, for infinitesimal positive loops around . In regions where the first derivative is not zero, holomorphic functions are conformal: they preserve angles and the shape (but not size) of small figures. Every holomorphic function is analytic. That is, a holomorphic function has derivatives of every order at each point in its domain, and it coincides with its own
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
at in a neighbourhood of . In fact, coincides with its Taylor series at in any disk centred at that point and lying within the domain of the function. From an algebraic point of view, the set of holomorphic functions on an open set is a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
and a
complex vector space 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 ...
. Additionally, the set of holomorphic functions in an open set is an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
if and only if the open set is connected. In fact, it is a
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
, with the seminorms being the suprema on
compact subset In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s. From a geometric perspective, a function is holomorphic at if and only if its
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
in a neighbourhood of is equal to for some continuous function . It follows from :\textstyle 0 = d^2 f = d(f^\prime dz) = df^\prime \wedge dz that is also proportional to , implying that the derivative is itself holomorphic and thus that is infinitely differentiable. Similarly, implies that any function that is holomorphic on the simply connected region is also integrable on . (For a path from to lying entirely in , define F_\gamma(z) = F_0 + \int_\gamma f\,dz; in light of the
Jordan curve theorem In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
and the
generalized Stokes' theorem A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
, is independent of the particular choice of path , and thus is a well-defined function on having and .)


Examples

All
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
functions in with complex
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s are
entire functions In complex analysis, an entire function, also called an integral function, is a complex-valued Function (mathematics), function that is holomorphic function, holomorphic on the whole complex plane. Typical examples of entire functions are polynomia ...
(holomorphic in the whole complex plane ), and so are the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
and the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
\cos = \tfrac12\bigl(\exp(iz) + \exp(-iz)\bigr) and \sin = -\tfrac12i\bigl(\exp(iz) - \exp(-iz)\bigr) (cf.
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
). The
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are used ...
of the
complex logarithm In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
function is holomorphic on the domain The
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
function can be defined as \sqrt = \exp\bigl(\tfrac12 \log z\bigr) and is therefore holomorphic wherever the logarithm is. The
reciprocal function In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
is holomorphic on (The reciprocal function, and any other
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
, is
meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
on .) As a consequence of the
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differ ...
, any real-valued holomorphic function must be constant. Therefore, the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, the
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
, the
real 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 a ...
and the
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 ...
are not holomorphic. Another typical example of a continuous function which is not holomorphic is 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 - ...
(The complex conjugate is
antiholomorphic In mathematics, antiholomorphic functions (also called antianalytic functionsEncyclopedia of Mathematics, Springer and The European Mathematical Society, https://encyclopediaofmath.org/wiki/Anti-holomorphic_function, As of 11 September 2020, This ...
.)


Several variables

The definition of a holomorphic function generalizes to several complex variables in a straightforward way. Let to be polydisk and also, denote an open subset of , and let . The function is analytic at a point in if there exists an open neighbourhood of in which is equal to a convergent power series in complex variables.Gunning and Rossi, ''Analytic Functions of Several Complex Variables'', p. 2. Define to be holomorphic if it is analytic at each point in its domain.
Osgood's lemma In mathematics, Osgood's lemma, introduced by , is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic In mathematics, a holomorphic function is a complex-valued function ...
shows (using the multivariate Cauchy integral formula) that, for a continuous function , this is equivalent to being holomorphic in each variable separately (meaning that if any coordinates are fixed, then the restriction of is a holomorphic function of the remaining coordinate). The much deeper
Hartogs' theorem In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of Function of several complex variables, several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. Mor ...
proves that the continuity assumption is unnecessary: is holomorphic if and only if it is holomorphic in each variable separately. More generally, a function of several complex variables that is
square integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
over every
compact subset In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions. Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex
Reinhardt domain The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
s, the simplest example of which is a
polydisk In the theory of functions of Function of several complex variables, several complex variables, a branch of mathematics, a polydisc is a Cartesian product of disc (mathematics), discs. More specifically, if we denote by D(z,r) the open set, ope ...
. However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called a
domain of holomorphy In mathematics, in the theory of functions of Function of several complex variables, 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 ...
. A complex differential -form is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero, .


Extension to functional analysis

The concept of a holomorphic function can be extended to the infinite-dimensional spaces of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. For instance, the Fréchet or
Gateaux derivative In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...
can be used to define a notion of a holomorphic function on a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
over the field of complex numbers.


See also

*
Antiderivative (complex analysis) In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function ''g'' is a function whose complex derivative is ''g''. More precisely, given an open set U in the complex plane and a function g:U\to \ ...
*
Antiholomorphic function In mathematics, antiholomorphic functions (also called antianalytic functionsEncyclopedia of Mathematics, Springer and The European Mathematical Society, https://encyclopediaofmath.org/wiki/Anti-holomorphic_function, As of 11 September 2020, This ...
*
Biholomorphy In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formal defini ...
* Holomorphic separability *
Meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
* Quadrature domains *
Harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
s *
Harmonic morphism In mathematics, a harmonic morphism is a (smooth) map \phi:(M^m,g)\to (N^n,h) between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class o ...
s *
Wirtinger derivatives In complex analysis of one and several complex variables, 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 sev ...


References


Further reading

*


External links

* {{Authority control Analytic functions