Holomorphic function: Difference between revisions

Line 1:

{{~~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}}

[[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a [[conformal map]] {{tmath|f}} (bottom).]]

[[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>~~.]]

[[File:Mapping f z equal 1 over z.gif|thumb|Mapping of the function {{tmath|1= f(z)={1}/{z} }}. The animation shows different {{tmath|z}} in blue color with the corresponding {{tmath|f(z)}} in red color. The point {{tmath|z}} and {{tmath|f(z)}} are shown in the {{tmath|1= \mathbb{C}\tilde = \mathbb{R}^2 }}. y-axis represents the imaginary part of the complex number of <math>z</math> and {{tmath| f(z) }}.]]

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]].

Line 23:

== 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.]]

[[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|\tfrac{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>

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>{{citation |last=Ahlfors |first=L. |author-link=Lars Ahlfors |title=Complex Analysis |edition=3rd |publisher=McGraw-Hill |date=1979 }}</ref>

:<math>f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{ z - z_0 }.</math>

: <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>

Line 41:

|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}}.

A function may be complex differentiable at a point but not holomorphic at this point. For example, the function {{tmath|1= \textstyle f(z) = \vert z \vert \vphantom{l}^2 = z \bar{z} }} ''is'' complex differentiable at {{tmath|0}}, but ''is not'' complex differentiable anywhere else, especially 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>

Line 53:

}} [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>

: <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

Line 67:

}}

</ref>

:<math>\frac{\partial f}{\partial\bar{z}} = 0,</math>

: <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}}.

Line 75:

|first2=S.A. |last2=Morris

|date=April 1978

|title=When is a function that satisfies the ~~Cauchy-Riemann~~ equations analytic?

|title=When is a function that satisfies the Cauchy–Riemann equations analytic?

|journal=[[The American Mathematical Monthly]]

|volume=85 |issue=4 |pages=246–256

Line 82:

</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 {{tmath| f(z) }} is a complex function that is complex differentiable about a point {{tmath|1= z = x + iy }} then (as we did earlier in the article) we can write {{tmath|1= f(z) = f(x+iy) = u(x,y) + i v(x,y) }} and then the complex derivative of the function can be written as {{tmath|1= f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} }}.<ref>

{{cite web

| last = Ponce Campuzano

Line 92:

| 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

}} </ref>

}}</ref>

== Terminology ==

Line 107:

| title = Applied and Computational Complex Analysis

| volume = 3

| place = New York - Chichester - Brisbane - Toronto - Singapore

| place = New York – Chichester – Brisbane – Toronto – Singapore

| publisher = [[John Wiley & Sons]]

| series = Wiley Classics Library

Line 137:

}}

</ref>

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

:<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}}:

[[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 {{tmath| D \equiv \{ z : \vert z - z_0 \vert \leq r \} }} 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>

:<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>

:<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>

:<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}}.

Line 163:

Line 156:

| year=1987

| title=Real and Complex Analysis

| publisher=~~McGraw–Hill~~ Book Co.

| publisher=McGraw-Hill Book Co.

| location=New York

| edition=3rd

Line 172:

Line 165:

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

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

:<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} }}.)

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 ==

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]].)

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]] {{tmath| \vert z \vert }}, 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 ==

Line 199:

Line 192:

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}}.

== Extension to functional analysis ==

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.

Line 237:

Line 230:

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

[[Category:Analytic functions]]

@@ Line 1: / Line 1: @@
-{{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}}
-{{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).]]
 {{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>.]]
+[[File:Mapping f z equal 1 over z.gif|thumb|Mapping of the function {{tmath|1= f(z)={1}/{z} }}. The animation shows different {{tmath|z}} in blue color with the corresponding {{tmath|f(z)}} in red color. The point {{tmath|z}} and {{tmath|f(z)}} are shown in the {{tmath|1= \mathbb{C}\tilde = \mathbb{R}^2 }}. y-axis represents the imaginary part of the complex number of <math>z</math> and {{tmath| f(z) }}.]]
 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]].
@@ Line 23: / Line 23: @@
 == 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.]]
+[[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|\tfrac{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>
+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>{{citation |last=Ahlfors |first=L. |author-link=Lars Ahlfors |title=Complex Analysis |edition=3rd |publisher=McGraw-Hill |date=1979 }}</ref>
-:<math>f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{ z - z_0 }.</math>
+: <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>
@@ Line 41: / Line 41: @@
 |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}}.
+A function may be complex differentiable at a point but not holomorphic at this point. For example, the function {{tmath|1= \textstyle f(z) = \vert z \vert \vphantom{l}^2 = z \bar{z} }} ''is'' complex differentiable at {{tmath|0}}, but ''is not'' complex differentiable anywhere else, especially 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>
@@ Line 53: / Line 53: @@
 }} [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>
+: <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
@@ Line 67: / Line 67: @@
 }}
 </ref>
-:<math>\frac{\partial f}{\partial\bar{z}} = 0,</math>
+: <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}}.
@@ Line 75: / Line 75: @@
   |first2=S.A. |last2=Morris
   |date=April 1978
-  |title=When is a function that satisfies the Cauchy-Riemann equations analytic?
+  |title=When is a function that satisfies the Cauchy–Riemann equations analytic?
   |journal=[[The American Mathematical Monthly]]
   |volume=85 |issue=4 |pages=246–256
@@ Line 82: / Line 82: @@
 </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 {{tmath| f(z) }} is a complex function that is complex differentiable about a point {{tmath|1= z = x + iy }} then (as we did earlier in the article) we can write {{tmath|1= f(z) = f(x+iy) = u(x,y) + i v(x,y) }} and then the complex derivative of the function can be written as {{tmath|1= f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} }}.<ref>
 {{cite web
   | last       = Ponce Campuzano
@@ Line 92: / Line 92: @@
   | 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
-}} </ref>
+}}</ref>
 == Terminology ==
@@ Line 107: / Line 107: @@
   | title = Applied and Computational Complex Analysis
   | volume = 3
-  | place = New York - Chichester - Brisbane - Toronto - Singapore
+  | place = New York – Chichester – Brisbane – Toronto – Singapore
   | publisher = [[John Wiley & Sons]]
   | series = Wiley Classics Library
@@ Line 137: / Line 137: @@
 }}
 </ref>
+: <math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math>
-:<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}}:
+[[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 {{tmath| D \equiv \{ z : \vert z - z_0 \vert \leq r \} }} 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>
-:<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>
-:<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>
-:<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}}.
@@ Line 163: / Line 156: @@
   | year=1987
   | title=Real and Complex Analysis
-  | publisher=McGraw–Hill Book Co.
+  | publisher=McGraw-Hill Book Co.
   | location=New York
   | edition=3rd
@@ Line 172: / Line 165: @@
 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
+: <math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math>
-:<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} }}.)
+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 ==
 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]].)
+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]] {{tmath| \vert z \vert }}, 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 ==
@@ Line 199: / Line 192: @@
 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}}.
 == 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.
@@ Line 237: / Line 230: @@
 * {{springer|title=Analytic function|id=p/a012240}}
-{{Authority control}}
+{{authority control}}
 [[Category:Analytic functions]]

Holomorphic function: Difference between revisions

Navigation menu

Search