Holomorphic function - Revision history

imported>AllCatsAreGrey: clean up, typo(s) fixed: ically- → ically

2025-10-13T01:28:12Z

clean up, typo(s) fixed: ically- → ically

← Previous revision		Revision as of 01:28, 13 October 2025
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	~~{{Use American English\|date = February 2019}}~~
	{{Short description\|Complex-differentiable (mathematical) function}}		{{Short description\|Complex-differentiable (mathematical) function}}
	{{for\|Zariski's theory of holomorphic functions on an algebraic variety\|formal holomorphic function}}		{{for\|Zariski's theory of holomorphic functions on an algebraic variety\|formal holomorphic function}}
	{{Redirect-distinguish\|Holomorphism\|Homomorphism}}		{{Redirect-distinguish\|Holomorphism\|Homomorphism}}
			{{Use American English\|date = February 2019}}
	[[Image:Conformal map.svg\|right\|thumb\|A rectangular grid (top) and its image under a [[conformal map]] {{tmath\|f}} (bottom).]]		[[Image:Conformal map.svg\|right\|thumb\|A rectangular grid (top) and its image under a [[conformal map]] {{tmath\|f}} (bottom).]]
	{{Complex analysis sidebar}}		{{Complex analysis sidebar}}
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	\|isbn=978-3-0346-0009-5 }}		\|isbn=978-3-0346-0009-5 }}
	</ref>		</ref>
	A function is ''holomorphic'' on some non-open set {{tmath\|A}} if it is holomorphic at every point of {{tmath\|A}}.		A function is ''holomorphic'' on some non-open set {{tmath\|A}} if it is holomorphic at every point of {{tmath\|A}}.

	A function may be complex differentiable at a point but not holomorphic at this point. For example, the function <math>\textstyle f(z) = \|z\|\vphantom{l}^2 = z\bar{z}</math> ''is'' complex differentiable at {{tmath\|0}}, but ''is not'' complex differentiable anywhere else, esp. including in no place close to {{tmath\|0}} (see the Cauchy–Riemann equations, below). So, it is ''not'' holomorphic at {{tmath\|0}}.		A function may be complex differentiable at a point but not holomorphic at this point. For example, the function <math>\textstyle f(z) = \|z\|\vphantom{l}^2 = z\bar{z}</math> ''is'' complex differentiable at {{tmath\|0}}, but ''is not'' complex differentiable anywhere else, esp. including in no place close to {{tmath\|0}} (see the Cauchy–Riemann equations, below). So, it is ''not'' holomorphic at {{tmath\|0}}.

	The relationship between real differentiability and complex differentiability is the following: If a complex function {{tmath\|1= f(x+ iy) = u(x,y) + i\,v(x, y)}} is holomorphic, then {{tmath\|u}} and {{tmath\|v}} have first partial derivatives with respect to {{tmath\|x}} and {{tmath\|y}}, and satisfy the [[Cauchy–Riemann equations]]:<ref name=Mark>		The relationship between real differentiability and complex differentiability is the following: If a complex function {{tmath\|1= f(x+ iy) = u(x,y) + i\,v(x, y)}} is holomorphic, then {{tmath\|u}} and {{tmath\|v}} have first partial derivatives with respect to {{tmath\|x}} and {{tmath\|y}}, and satisfy the [[Cauchy–Riemann equations]]:<ref name=Mark>
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	</ref>		</ref>

	An immediate useful consequence of the Cauchy Riemann Equations above is that the complex derivative can be defined explicitly in terms of real partial derivatives. If <math> f(z) </math> is a complex function that is complex differentiable about a point <math> z = x+ iy </math> then (as we did earlier in the article) we can write <math> f(z) = f(x+iy) = u(x,y) + i v(x,y)</math> and then the complex derivative of the function can be written as <math> f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} </math> <ref>		An immediate useful consequence of the Cauchy Riemann Equations above is that the complex derivative can be defined explicitly in terms of real partial derivatives. If <math> f(z) </math> is a complex function that is complex differentiable about a point <math> z = x+ iy </math> then (as we did earlier in the article) we can write <math> f(z) = f(x+iy) = u(x,y) + i v(x,y)</math> and then the complex derivative of the function can be written as <math> f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} </math> <ref>
	{{cite web		{{cite web
	\| last = Ponce Campuzano		\| last = Ponce Campuzano
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	\| url = https://math.libretexts.org/Bookshelves/Analysis/Complex_Analysis_-_A_Visual_and_Interactive_Introduction_(Ponce_Campuzano)/02%3A_Chapter_2/2.03%3A_Complex_Differentiation		\| url = https://math.libretexts.org/Bookshelves/Analysis/Complex_Analysis_-_A_Visual_and_Interactive_Introduction_(Ponce_Campuzano)/02%3A_Chapter_2/2.03%3A_Complex_Differentiation
	\| access-date= 15 June 2025		\| access-date= 15 June 2025
	}} </ref>		}}</ref>

	== Terminology ==		== Terminology ==
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	Every [[holomorphic functions are analytic\|holomorphic function is analytic]]. That is, a holomorphic function {{tmath\|f}} has derivatives of every order at each point {{tmath\|a}} in its domain, and it coincides with its own [[Taylor series]] at {{tmath\|a}} in a neighbourhood of {{tmath\|a}}. In fact, {{tmath\|f}} coincides with its Taylor series at {{tmath\|a}} in any disk centred at that point and lying within the domain of the function.		Every [[holomorphic functions are analytic\|holomorphic function is analytic]]. That is, a holomorphic function {{tmath\|f}} has derivatives of every order at each point {{tmath\|a}} in its domain, and it coincides with its own [[Taylor series]] at {{tmath\|a}} in a neighbourhood of {{tmath\|a}}. In fact, {{tmath\|f}} coincides with its Taylor series at {{tmath\|a}} 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]] and a [[complex vector space]]. Additionally, the set of holomorphic functions in an open set {{tmath\|U}} is an [[integral domain]] [[if and only if]] the open set {{tmath\|U}} is connected. <ref name=Gunning/> In fact, it is a [[locally convex topological vector space]], with the [[norm (mathematics)\|seminorms]] being the [[suprema]] on [[compact subset]]s.		From an algebraic point of view, the set of holomorphic functions on an open set is a [[commutative ring]] and a [[complex vector space]]. Additionally, the set of holomorphic functions in an open set {{tmath\|U}} is an [[integral domain]] [[if and only if]] the open set {{tmath\|U}} is connected.<ref name=Gunning/> In fact, it is a [[locally convex topological vector space]], with the [[norm (mathematics)\|seminorms]] being the [[suprema]] on [[compact subset]]s.

	From a geometric perspective, a function {{tmath\|f}} is holomorphic at {{tmath\|z_0}} if and only if its [[exterior derivative]] {{tmath\|\mathrm{d}f}} in a neighbourhood {{tmath\|U}} of {{tmath\|z_0}} is equal to {{tmath\| f'(z)\,\mathrm{d}z}} for some continuous function {{tmath\|f'}}. It follows from		From a geometric perspective, a function {{tmath\|f}} is holomorphic at {{tmath\|z_0}} if and only if its [[exterior derivative]] {{tmath\|\mathrm{d}f}} in a neighbourhood {{tmath\|U}} of {{tmath\|z_0}} is equal to {{tmath\| f'(z)\,\mathrm{d}z}} for some continuous function {{tmath\|f'}}. It follows from
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	that {{tmath\|\mathrm{d}f'}} is also proportional to {{tmath\|\mathrm{d}z}}, implying that the derivative {{tmath\|\mathrm{d}f'}} is itself holomorphic and thus that {{tmath\|f}} is infinitely differentiable. Similarly, {{tmath\|1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0}} implies that any function {{tmath\|f}} that is holomorphic on the simply connected region {{tmath\|U}} is also integrable on {{tmath\|U}}.		that {{tmath\|\mathrm{d}f'}} is also proportional to {{tmath\|\mathrm{d}z}}, implying that the derivative {{tmath\|\mathrm{d}f'}} is itself holomorphic and thus that {{tmath\|f}} is infinitely differentiable. Similarly, {{tmath\|1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0}} implies that any function {{tmath\|f}} that is holomorphic on the simply connected region {{tmath\|U}} is also integrable on {{tmath\|U}}.

	(For a path {{tmath\|\gamma}} from {{tmath\|z_0}} to {{tmath\|z}} lying entirely in {{tmath\|U}}, define {{tmath\|1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z }}~~; in~~ light of the [[Jordan curve theorem]] and the [[Stokes' theorem\|generalized Stokes' theorem]], {{tmath\|F_\gamma(z)}} is independent of the particular choice of path {{tmath\|\gamma}}, and thus {{tmath\|F(z)}} is a well-defined function on {{tmath\|U}} having {{tmath\|1= \mathrm{d}F = f\,\mathrm{d}z}} or {{tmath\|1= f = ~~\frac{~~\mathrm{d}F}{\mathrm{d}z} }}.)		For a path {{tmath\|\gamma}} from {{tmath\|z_0}} to {{tmath\|z}} lying entirely in {{tmath\|U}}, define

			:{{tmath\|1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z.}}
			In light of the [[Jordan curve theorem]] and the [[Stokes' theorem\|generalized Stokes' theorem]], {{tmath\|F_\gamma(z)}} is independent of the particular choice of path {{tmath\|\gamma}}, and thus {{tmath\|F(z)}} is a well-defined function on {{tmath\|U}} having {{tmath\|1= \mathrm{d}F = f\,\mathrm{d}z}}, or equivalently {{tmath\|1= f = \mathrm{d}F/\mathrm{d}z }}.

	== Examples ==		== Examples ==
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	More generally, a function of several complex variables that is [[square integrable]] over every [[compact set\|compact subset]] of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.		More generally, a function of several complex variables that is [[square integrable]] over every [[compact set\|compact subset]] 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]]s, the simplest example of which is a [[polydisk]]. 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]].		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]]s, the simplest example of which is a [[polydisk]]. 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]].

	A [[complex differential form#Holomorphic forms\|complex differential {{tmath\|(p,0)}}-form]] {{tmath\|\alpha}} is holomorphic if and only if its antiholomorphic [[Complex differential form#The Dolbeault operators\|Dolbeault derivative]] is zero: {{tmath\|1= \bar{\partial}\alpha = 0}}.		A [[complex differential form#Holomorphic forms\|complex differential {{tmath\|(p,0)}}-form]] {{tmath\|\alpha}} is holomorphic if and only if its antiholomorphic [[Complex differential form#The Dolbeault operators\|Dolbeault derivative]] is zero: {{tmath\|1= \bar{\partial}\alpha = 0}}.

	== Extension to functional analysis ==		== Extension to functional analysis ==
	{{Main ~~article~~\|infinite-dimensional holomorphy}}		{{Main\|infinite-dimensional holomorphy}}
	The concept of a holomorphic function can be extended to the infinite-dimensional spaces of [[functional analysis]]. For instance, the [[Fréchet derivative\|Fréchet]] or [[Gateaux derivative]] can be used to define a notion of a holomorphic function on a [[Banach space]] over the field of complex numbers.		The concept of a holomorphic function can be extended to the infinite-dimensional spaces of [[functional analysis]]. For instance, the [[Fréchet derivative\|Fréchet]] or [[Gateaux derivative]] can be used to define a notion of a holomorphic function on a [[Banach space]] over the field of complex numbers.

imported>Frogeyedpeas: /* Definition */ there is literally no definition of complex derivative on the definitions section. The cauchy riemann equations merely allude to the existence of an explicit definition in terms of real partial derivatives but this does a massive disservice to readers coming here to actually learn WHAT the complex derivative IS in terms of simpler things to understand like real partial derivatives.

2025-06-15T21:26:27Z

Definition: there is literally no definition of complex derivative on the definitions section. The cauchy riemann equations merely allude to the existence of an explicit definition in terms of real partial derivatives but this does a massive disservice to readers coming here to actually learn WHAT the complex derivative IS in terms of simpler things to understand like real partial derivatives.

@@ Line 81: / Line 81: @@
 }}
 </ref>
 == Terminology ==

imported>Riffraff11235: The "tmath" environment used in this article doesn't seem to play nicely with in-line math expressions that include curly brackets, such as the definition of the open disc D in this section. Replacing the "tmath" wrapper with " $" seems to have done the trick.$

2025-05-11T16:37:23Z

The "tmath" environment used in this article doesn't seem to play nicely with in-line math expressions that include curly brackets, such as the definition of the open disc D in this section. Replacing the "tmath" wrapper with "<math>" seems to have done the trick.

New page

{{Use American English|date = February 2019}}
{{Short description|Complex-differentiable (mathematical) function}}
{{for|Zariski's theory of holomorphic functions on an algebraic variety|formal holomorphic function}}
{{Redirect-distinguish|Holomorphism|Homomorphism}}
[[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a [[conformal map]] {{tmath|f}} (bottom).]]
{{Complex analysis sidebar}}
[[File:Mapping f z equal 1 over z.gif|thumb|Mapping of the function <math>f(z)=\frac{1}{z}</math>. The animation shows different <math>z</math> in blue color with the corresponding <math>f(z)</math> in red color. The point <math>z</math> and <math>f(z)</math> are shown in the <math>\mathbb{C}\tilde{=}\mathbb{R}^2</math>. y-axis represents the imaginary part of the complex number of <math>z</math> and <math>f(z)</math>.]]

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 differentiable]] in a [[neighbourhood (mathematics)|neighbourhood]] of each point in a [[domain (mathematical analysis)|domain]] in [[Function of several complex variables#The complex coordinate space|complex coordinate space]] {{tmath|\C^n}}. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is [[infinitely differentiable]] and locally equal to its own [[Taylor series]] (is ''[[analytic function|analytic]]''). Holomorphic functions are the central objects of study in [[complex analysis]].

Though the term ''[[analytic function]]'' 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 of a function|domain]]. That all holomorphic functions are complex analytic functions, and vice versa, is a [[Holomorphic functions are analytic|major theorem in complex analysis]].<ref>
{{cite encyclopedia
|title=Analytic functions of one complex variable
|year=2015
|encyclopedia=Encyclopedia of Mathematics
|publisher=European Mathematical Society / Springer
|url=https://www.encyclopediaofmath.org/index.php/Analytic_function#Analytic_functions_of_one_complex_variable
|via=encyclopediaofmath.org
}}
</ref>

Holomorphic functions are also sometimes referred to as ''regular functions''.<ref>{{SpringerEOM|title=Analytic function|access-date=February 26, 2021}}</ref> A holomorphic function whose domain is the whole [[complex plane]] is called an [[entire function]]. The phrase "holomorphic at a point {{tmath|z_0}}" means not just differentiable at {{tmath|z_0}}, but differentiable everywhere within some close neighbourhood of {{tmath|z_0}} in the complex plane.

== Definition ==
[[File:Non-holomorphic complex conjugate.svg|thumb|The function {{tmath|1=f(z) = \bar{z} }} is not complex differentiable at zero, because as shown above, the value of {{tmath|\frac{f(z) - f(0)}{z - 0} }} varies depending on the direction from which zero is approached. On the real axis only, {{tmath|f}} equals the function {{tmath|1=g(z) = z}} and the limit is {{tmath|1}}, while along the imaginary axis only, {{tmath|f}} equals the different function {{tmath|1=h(z) = -z}} and the limit is {{tmath|-1}}. Other directions yield yet other limits.]]
Given a complex-valued function {{tmath|f}} of a single complex variable, the '''derivative''' of {{tmath|f}} at a point {{tmath|z_0}} in its domain is defined as the [[limit of a function|limit]]<ref>[[Lars Ahlfors|Ahlfors, L.]], ''Complex Analysis, 3 ed.'' (McGraw-Hill, 1979).</ref>
:<math>f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{ z - z_0 }.</math>

This is the same definition as for the [[derivative]] of a [[real function]], except that all quantities are complex. In particular, the limit is taken as the complex number {{tmath|z}} tends to {{tmath|z_0}}, and this means that the same value is obtained for any sequence of complex values for {{tmath|z}} that tends to {{tmath|z_0}}. If the limit exists, {{tmath|f}} is said to be '''complex differentiable''' at {{tmath|z_0}}. This concept of complex differentiability shares several properties with [[derivative|real differentiability]]: It is [[linear transformation|linear]] and obeys the [[product rule]], [[quotient rule]], and [[chain rule]].<ref>{{cite book |author-link=Peter Henrici (mathematician) |last=Henrici |first=P. |title=Applied and Computational Complex Analysis |publisher=Wiley |year=1986 |orig-year=1974, 1977 }} Three volumes, publ.: 1974, 1977, 1986.</ref>

A function is '''holomorphic''' on an [[open set]] {{tmath|U}} if it is ''complex differentiable'' at ''every'' point of {{tmath|U}}. A function {{tmath|f}} is ''holomorphic'' at a point {{tmath|z_0}} if it is holomorphic on some [[neighbourhood (mathematics)|neighbourhood]] of {{tmath|z_0}}.<ref>
{{cite book
|first1=Peter |last1=Ebenfelt |first2=Norbert |last2=Hungerbühler
|first3=Joseph J. |last3=Kohn |first4=Ngaiming |last4=Mok
|first5=Emil J. |last5=Straube
|year=2011
|url=https://books.google.com/books?id=3GeUgafFRgMC&q=holomorphic |via=Google
|title=Complex Analysis
|publisher=Springer
|series=Science & Business Media
|isbn=978-3-0346-0009-5 }}
</ref>
A function is ''holomorphic'' on some non-open set {{tmath|A}} if it is holomorphic at every point of {{tmath|A}}.

A function may be complex differentiable at a point but not holomorphic at this point. For example, the function <math>\textstyle f(z) = |z|\vphantom{l}^2 = z\bar{z}</math> ''is'' complex differentiable at {{tmath|0}}, but ''is not'' complex differentiable anywhere else, esp. including in no place close to {{tmath|0}} (see the Cauchy–Riemann equations, below). So, it is ''not'' holomorphic at {{tmath|0}}.

The relationship between real differentiability and complex differentiability is the following: If a complex function {{tmath|1= f(x+ iy) = u(x,y) + i\,v(x, y)}} is holomorphic, then {{tmath|u}} and {{tmath|v}} have first partial derivatives with respect to {{tmath|x}} and {{tmath|y}}, and satisfy the [[Cauchy–Riemann equations]]:<ref name=Mark>
{{cite book
|last=Markushevich |first=A.I.
|year=1965
|title=Theory of Functions of a Complex Variable
|publisher=Prentice-Hall
}} [In three volumes.]
</ref>
:<math>\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \mbox{and} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\,</math>
or, equivalently, the [[Wirtinger derivative]] of {{tmath|f}} with respect to {{tmath|\bar z}}, the [[complex conjugate]] of {{tmath|z}}, is zero:<ref name=Gunning>
{{cite book
|last1 = Gunning |first1 = Robert C. |author1-link = Robert Gunning (mathematician)
|last2 = Rossi |first2 = Hugo
|year = 1965
|title = Analytic Functions of Several Complex Variables
|series = Modern Analysis
|place = Englewood Cliffs, NJ
|publisher = [[Prentice-Hall]]
|mr = 0180696 |zbl = 0141.08601 |isbn = 9780821869536
|url = https://books.google.com/books?id=L0zJmamx5AAC |via=Google
}}
</ref>
:<math>\frac{\partial f}{\partial\bar{z}} = 0,</math>
which is to say that, roughly, {{tmath|f}} is functionally independent from {{tmath|\bar z}}, the complex conjugate of {{tmath|z}}.

If continuity is not given, the converse is not necessarily true. A simple converse is that if {{tmath|u}} and {{tmath|v}} have ''continuous'' first partial derivatives and satisfy the Cauchy–Riemann equations, then {{tmath|f}} is holomorphic. A more satisfying converse, which is much harder to prove, is the [[Looman–Menchoff theorem]]: if {{tmath|f}} is continuous, {{tmath|u}} and {{tmath|v}} have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then {{tmath|f}} is holomorphic.<ref>
{{cite journal
|first1=J.D. |last1=Gray
|first2=S.A. |last2=Morris
|date=April 1978
|title=When is a function that satisfies the Cauchy-Riemann equations analytic?
|journal=[[The American Mathematical Monthly]]
|volume=85 |issue=4 |pages=246–256
|jstor=2321164 |doi=10.2307/2321164
}}
</ref>

== Terminology ==
The term ''holomorphic'' was introduced in 1875 by [[Charles Auguste Briot|Charles Briot]] and [[Jean-Claude Bouquet]], two of [[Augustin-Louis Cauchy]]'s students, and derives from the Greek [[wikt:ὅλος|ὅλος]] (''hólos'') meaning "whole", and [[wikt:μορφή|μορφή]] (''morphḗ'') meaning "form" or "appearance" or "type", in contrast to the term ''[[meromorphic function|meromorphic]]'' derived from [[wikt:μέρος|μέρος]] (''méros'') meaning "part". A holomorphic function resembles an [[entire function]] ("whole") in a [[domain (mathematical analysis)|domain]] of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated [[Zeros and poles|poles]]), resembles a rational fraction ("part") of entire functions in a domain of the complex plane.<ref>The original French terms were ''holomorphe'' and ''méromorphe''. {{pb}}
{{cite book |authorlink1= Charles Auguste Briot |last1=Briot |first1=Charles Auguste |authorlink2=Jean-Claude Bouquet |last2=Bouquet |first2=Jean-Claude |date=1875 |title=Théorie des fonctions elliptiques |edition=2nd |publisher=Gauthier-Villars |chapter=§15 fonctions holomorphes |chapter-url=https://archive.org/details/thoriedesfonct00briouoft/page/14/ |pages=14–15 |quote=Lorsqu'une fonction est continue, monotrope, et a une dérivée, quand la variable se meut dans une certaine partie du plan, nous dirons qu'elle est ''holomorphe'' dans cette partie du plan. Nous indiquons par cette dénomination qu'elle est semblable aux fonctions entières qui jouissent de ces propriétés dans toute l'étendue du plan. [...] ¶ Une fraction rationnelle admet comme pôles les racines du dénominateur; c'est une fonction holomorphe dans toute partie du plan qui ne contient aucun de ses pôles. ¶ Lorsqu'une fonction est holomorphe dans une partie du plan, excepté en certains pôles, nous dirons qu'elle est ''méromorphe'' dans cette partie du plan, c'est-à-dire semblable aux fractions rationnelles. |trans-quote=When a function is continuous, [[Monodromy|monotropic]], and has a derivative, when the variable moves in a certain part of the [complex] plane, we say that it is ''holomorphic'' in that part of the plane. We mean by this name that it resembles [[entire function]]s which enjoy these properties in the full extent of the plane. [...] ¶ A rational fraction admits as [[zeros and poles|poles]] the [[zeros and poles|roots]] of the denominator; it is a holomorphic function in all that part of the plane which does not contain any poles. ¶ When a function is holomorphic in part of the plane, except at certain poles, we say that it is ''meromorphic'' in that part of the plane, that is to say it resembles rational fractions.}} {{pb}} {{cite book |authorlink1=James Harkness (mathematician) |first1=James |last1=Harkness |authorlink2=Frank Morley |first2=Frank |last2=Morley |date=1893 |chapter=5. Integration |chapter-url=https://archive.org/details/treatiseontheory00harkrich/page/n176/ |title=A Treatise on the Theory of Functions |publisher=Macmillan |page=161}}</ref> Cauchy had instead used the term ''synectic''.<ref>Briot & Bouquet had previously also adopted Cauchy’s term ''synectic'' (''synectique'' in French), in the 1859 first edition of their book. {{pb}} {{cite book |authorlink1= Charles Auguste Briot |last1=Briot |first1=Charles Auguste |authorlink2=Jean-Claude Bouquet |last2=Bouquet |first2=Jean-Claude |date=1859 |title= Théorie des fonctions doublement périodiques |publisher= Mallet-Bachelier |chapter=§10 |chapter-url=https://archive.org/details/fonctsdoublement00briorich/page/n37/ |page=11 }}</ref>

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.<ref>
{{cite book
| last = Henrici | first = Peter | author-link = Peter Henrici (mathematician)
| year = 1993 | orig-year = 1986
| title = Applied and Computational Complex Analysis
| volume = 3
| place = New York - Chichester - Brisbane - Toronto - Singapore
| publisher = [[John Wiley & Sons]]
| series = Wiley Classics Library
| edition = Reprint
| mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1
| url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google
}}
</ref> That is, if functions {{tmath|f}} and {{tmath|g}} are holomorphic in a domain {{tmath|U}}, then so are {{tmath|f+g}}, {{tmath|f-g}}, {{tmath| fg}}, and {{tmath|f \circ g}}. Furthermore, {{tmath|f/g }} is holomorphic if {{tmath|g}} has no zeros in {{tmath|U}}; otherwise it is [[meromorphic]].

If one identifies {{tmath|\C}} with the real [[plane (geometry)|plane]] {{tmath|\textstyle \R^2}}, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[Cauchy–Riemann equations]], a set of two [[partial differential equation]]s.<ref name=Mark/>

Every holomorphic function can be separated into its real and imaginary parts {{tmath|1=f(x + iy) = u(x, y) + i\,v(x,y)}}, and each of these is a [[harmonic function]] on {{tmath|\textstyle \R^2}} (each satisfies [[Laplace's equation]] {{tmath|1=\textstyle \nabla^2 u = \nabla^2 v = 0}}), with {{tmath|v}} the [[harmonic conjugate]] of {{tmath|u}}.<ref>
{{cite book
|first=L.C. |last=Evans |author-link=Lawrence C. Evans
|year=1998
|title=Partial Differential Equations
|publisher=American Mathematical Society
}}
</ref>
Conversely, every harmonic function {{tmath|u(x, y)}} on a [[Simply connected space|simply connected]] domain {{tmath|\textstyle \Omega \subset \R^2}} is the real part of a holomorphic function: If {{tmath|v}} is the harmonic conjugate of {{tmath|u}}, unique up to a constant, then {{tmath|1=f(x + iy) = u(x, y) + i\,v(x, y)}} is holomorphic.

[[Cauchy's integral theorem]] implies that the [[contour integral]] of every holomorphic function along a [[loop (topology)|loop]] vanishes:<ref name=Lang>
{{cite book
|first = Serge |last = Lang | author-link = Serge Lang
| year = 2003
| title = Complex Analysis
| series = Springer Verlag GTM
| publisher = [[Springer Verlag]]
}}
</ref>

:<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math>

Here {{tmath|\gamma}} is a [[rectifiable path]] in a simply connected [[domain (mathematical analysis)|complex domain]] {{tmath|U \subset \C}} whose start point is equal to its end point, and {{tmath|f \colon U \to \C}} is a holomorphic function.

[[Cauchy's integral formula]] states that every function holomorphic inside a [[disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose {{tmath|U \subset \C}} is a complex domain, {{tmath|f\colon U \to \C}} is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \leq r \} </math> is [[neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in {{tmath|U}}. Let {{tmath|\gamma}} be the circle forming the [[boundary (topology)|boundary]] of {{tmath|D}}. Then for every {{tmath|a}} in the [[interior (topology)|interior]] of {{tmath|D}}:

:<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math>

where the contour integral is taken [[curve orientation|counter-clockwise]].

The derivative {{tmath|{f'}(a)}} can be written as a contour integral<ref name=Lang /> using [[Cauchy's differentiation formula]]:

:<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math>

for any simple loop positively winding once around {{tmath|a}}, and

:<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math>

for [[infinitesimal]] positive loops {{tmath|\gamma}} around {{tmath|a}}.

In regions where the first derivative is not zero, holomorphic functions are [[conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref>
{{cite book
| last =Rudin | first =Walter | author-link = Walter Rudin
| year=1987
| title=Real and Complex Analysis
| publisher=McGraw–Hill Book Co.
| location=New York
| edition=3rd
| isbn=978-0-07-054234-1 | mr=924157
}}
</ref>

Every [[holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function {{tmath|f}} has derivatives of every order at each point {{tmath|a}} in its domain, and it coincides with its own [[Taylor series]] at {{tmath|a}} in a neighbourhood of {{tmath|a}}. In fact, {{tmath|f}} coincides with its Taylor series at {{tmath|a}} 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]] and a [[complex vector space]]. Additionally, the set of holomorphic functions in an open set {{tmath|U}} is an [[integral domain]] [[if and only if]] the open set {{tmath|U}} is connected. <ref name=Gunning/> In fact, it is a [[locally convex topological vector space]], with the [[norm (mathematics)|seminorms]] being the [[suprema]] on [[compact subset]]s.

From a geometric perspective, a function {{tmath|f}} is holomorphic at {{tmath|z_0}} if and only if its [[exterior derivative]] {{tmath|\mathrm{d}f}} in a neighbourhood {{tmath|U}} of {{tmath|z_0}} is equal to {{tmath| f'(z)\,\mathrm{d}z}} for some continuous function {{tmath|f'}}. It follows from

:<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math>

that {{tmath|\mathrm{d}f'}} is also proportional to {{tmath|\mathrm{d}z}}, implying that the derivative {{tmath|\mathrm{d}f'}} is itself holomorphic and thus that {{tmath|f}} is infinitely differentiable. Similarly, {{tmath|1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0}} implies that any function {{tmath|f}} that is holomorphic on the simply connected region {{tmath|U}} is also integrable on {{tmath|U}}.

(For a path {{tmath|\gamma}} from {{tmath|z_0}} to {{tmath|z}} lying entirely in {{tmath|U}}, define {{tmath|1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z }}; in light of the [[Jordan curve theorem]] and the [[Stokes' theorem|generalized Stokes' theorem]], {{tmath|F_\gamma(z)}} is independent of the particular choice of path {{tmath|\gamma}}, and thus {{tmath|F(z)}} is a well-defined function on {{tmath|U}} having {{tmath|1= \mathrm{d}F = f\,\mathrm{d}z}} or {{tmath|1= f = \frac{\mathrm{d}F}{\mathrm{d}z} }}.)

== Examples ==
All [[polynomial]] functions in {{tmath|z}} with complex [[coefficient]]s are [[entire function]]s (holomorphic in the whole complex plane {{tmath|\C}}), and so are the [[exponential function#Complex plane|exponential function]] {{tmath|\exp z}} and the [[trigonometric functions]] {{tmath|1= \cos{z} = \tfrac{1}{2} \bigl( \exp(+iz) + \exp(-iz)\bigr)}} and {{tmath|1= \sin{z} = -\tfrac{1}{2} i \bigl(\exp(+iz) - \exp(-iz)\bigr)}} (cf. [[Euler's formula]]). The [[principal branch]] of the [[complex logarithm]] function {{tmath|\log z}} is holomorphic on the domain {{tmath|\C \smallsetminus \{ z \in \R : z \le 0\} }}. The [[square root#Principal square root of a complex number|square root]] function can be defined as {{tmath|\sqrt{z} \equiv \exp \bigl(\tfrac{1}{2} \log z\bigr) }} and is therefore holomorphic wherever the logarithm {{tmath|\log z}} is. The [[multiplicative inverse#Complex numbers|reciprocal function]] {{tmath|\tfrac{1}{z} }} is holomorphic on {{tmath| \C \smallsetminus \{ 0 \} }}. (The reciprocal function, and any other [[rational function]], is [[meromorphic function|meromorphic]] on {{tmath|\C}}.)

As a consequence of the [[Cauchy–Riemann equations]], any real-valued holomorphic function must be [[constant function|constant]]. Therefore, the [[absolute value#Complex numbers|absolute value]] {{nobr|<math>|z|</math>,}} the [[argument (complex analysis)|argument]] {{tmath|\arg z}}, the [[Complex number#Notation|real part]] {{tmath|\operatorname{Re}(z)}} and the [[Complex number#Notation|imaginary part]] {{tmath|\operatorname{Im}(z)}} are not holomorphic. Another typical example of a [[continuous function]] which is not holomorphic is the [[complex conjugate]] {{tmath|\bar z.}} (The complex conjugate is [[antiholomorphic function|antiholomorphic]].)

== Several variables ==
The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function {{tmath|f \colon ( z_1, z_2, \ldots, z_n ) \mapsto f( z_1, z_2, \ldots, z_n ) }} in {{tmath|n}} complex variables is '''analytic''' at a point {{tmath|p}} if there exists a neighbourhood of {{tmath|p}} in which {{tmath|f}} is equal to a convergent power series in {{tmath|n}} complex variables;<ref>
{{cite book
|last1=Gunning |last2=Rossi |name-list-style=and
|title=Analytic Functions of Several Complex Variables
|page=2
}}
</ref>
the function {{tmath|f}} is '''holomorphic''' in an open subset {{tmath|U}} of {{tmath|\C^n}} if it is analytic at each point in {{tmath|U}}. [[Osgood's lemma]] shows (using the multivariate Cauchy integral formula) that, for a continuous function {{tmath|f}}, this is equivalent to {{tmath|f}} being holomorphic in each variable separately (meaning that if any {{tmath|n-1}} coordinates are fixed, then the restriction of {{tmath|f}} is a holomorphic function of the remaining coordinate). The much deeper [[Hartogs' theorem]] proves that the continuity assumption is unnecessary: {{tmath|f}} 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]] over every [[compact set|compact subset]] 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]]s, the simplest example of which is a [[polydisk]]. 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]].

A [[complex differential form#Holomorphic forms|complex differential {{tmath|(p,0)}}-form]] {{tmath|\alpha}} is holomorphic if and only if its antiholomorphic [[Complex differential form#The Dolbeault operators|Dolbeault derivative]] is zero: {{tmath|1= \bar{\partial}\alpha = 0}}.

== Extension to functional analysis ==
{{Main article|infinite-dimensional holomorphy}}
The concept of a holomorphic function can be extended to the infinite-dimensional spaces of [[functional analysis]]. For instance, the [[Fréchet derivative|Fréchet]] or [[Gateaux derivative]] can be used to define a notion of a holomorphic function on a [[Banach space]] over the field of complex numbers.

== See also ==
{{div col begin|colwidth=18em}}
* [[Antiderivative (complex analysis)]]
* [[Antiholomorphic function]]
* [[Biholomorphy]]
* [[Cauchy's estimate]]
* [[Harmonic map]]s
* [[Harmonic morphism]]s
* [[Holomorphic separability]]
* [[Meromorphic function]]
* [[Quadrature domains]]
* [[Wirtinger derivatives]]
{{div col end}}

== References ==
{{reflist|25em}}

== Further reading ==
* {{cite book
|last=Blakey |first=Joseph
|year=1958
|title=University Mathematics |edition=2nd
|publisher=Blackie and Sons
|location=London, UK
|oclc=2370110
}}

== External links ==
* {{springer|title=Analytic function|id=p/a012240}}

{{Authority control}}

[[Category:Analytic functions]]

Holomorphic function - Revision history

imported>AllCatsAreGrey: clean up, typo(s) fixed: ically- → ically

imported>Riffraff11235: The "tmath" environment used in this article doesn't seem to play nicely with in-line math expressions that include curly brackets, such as the definition of the open disc D in this section. Replacing the "tmath" wrapper with "" seems to have done the trick.

imported>Riffraff11235: The "tmath" environment used in this article doesn't seem to play nicely with in-line math expressions that include curly brackets, such as the definition of the open disc D in this section. Replacing the "tmath" wrapper with " $" seems to have done the trick.$