Holomorphic function: Difference between revisions
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. |
imported>GroundclothDilemma →Definition: C/e |
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{{ | {{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}} | ||
{{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}} | ||
[[File:Mapping f z equal 1 over z.gif|thumb|Mapping of the function | [[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]]. | 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]]. | ||
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== Definition == | == 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|\ | [[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> | 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> | 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> | ||
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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 | 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> | 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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}} [In three volumes.] | }} [In three volumes.] | ||
</ref> | </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> | 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 | {{cite book | ||
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}} | }} | ||
</ref> | </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}}. | which is to say that, roughly, {{tmath|f}} is functionally independent from {{tmath|\bar z}}, the complex conjugate of {{tmath|z}}. | ||
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|first2=S.A. |last2=Morris | |first2=S.A. |last2=Morris | ||
|date=April 1978 | |date=April 1978 | ||
|title=When is a function that satisfies the | |title=When is a function that satisfies the Cauchy–Riemann equations analytic? | ||
|journal=[[The American Mathematical Monthly]] | |journal=[[The American Mathematical Monthly]] | ||
|volume=85 |issue=4 |pages=246–256 | |volume=85 |issue=4 |pages=246–256 | ||
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</ref> | </ref> | ||
An immediate useful consequence of the | 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 | {{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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| title = Applied and Computational Complex Analysis | | title = Applied and Computational Complex Analysis | ||
| volume = 3 | | volume = 3 | ||
| place = New York | | place = New York – Chichester – Brisbane – Toronto – Singapore | ||
| publisher = [[John Wiley & Sons]] | | publisher = [[John Wiley & Sons]] | ||
| series = Wiley Classics Library | | series = Wiley Classics Library | ||
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}} | }} | ||
</ref> | </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. | 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 | [[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]]. | 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]]: | 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 | 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}}. | for [[infinitesimal]] positive loops {{tmath|\gamma}} around {{tmath|a}}. | ||
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| year=1987 | | year=1987 | ||
| title=Real and Complex Analysis | | title=Real and Complex Analysis | ||
| publisher= | | publisher=McGraw-Hill Book Co. | ||
| location=New York | | location=New York | ||
| edition=3rd | | edition=3rd | ||
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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 | ||
: <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}}. | 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 equivalently {{tmath|1= f = \mathrm{d}F/\mathrm{d}z }}. | |||
== Examples == | == 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}}.) | 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]] {{ | 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 == | == Several variables == | ||
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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 | 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|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. | ||
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* {{springer|title=Analytic function|id=p/a012240}} | * {{springer|title=Analytic function|id=p/a012240}} | ||
{{ | {{authority control}} | ||
[[Category:Analytic functions]] | [[Category:Analytic functions]] | ||
Latest revision as of 12:35, 13 May 2026
Template:Complex analysis sidebar
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 Template:Tmath. 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). 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. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.[1]
Holomorphic functions are also sometimes referred to as regular functions.[2] A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point Template:Tmath" means not just differentiable at Template:Tmath, but differentiable everywhere within some close neighbourhood of Template:Tmath in the complex plane.
Definition
[edit | edit source]
Given a complex-valued function Template:Tmath of a single complex variable, the derivative of Template:Tmath at a point Template:Tmath in its domain is defined as the limit[3]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{ z - z_0 }.}
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 Template:Tmath tends to Template:Tmath, and this means that the same value is obtained for any sequence of complex values for Template:Tmath that tends to Template:Tmath. If the limit exists, Template:Tmath is said to be complex differentiable at Template:Tmath. This concept of complex differentiability shares several properties with real differentiability: It is linear and obeys the product rule, quotient rule, and chain rule.[4]
A function is holomorphic on an open set Template:Tmath if it is complex differentiable at every point of Template:Tmath. A function Template:Tmath is holomorphic at a point Template:Tmath if it is holomorphic on some neighbourhood of Template:Tmath.[5] A function is holomorphic on some non-open set Template:Tmath if it is holomorphic at every point of Template:Tmath.
A function may be complex differentiable at a point but not holomorphic at this point. For example, the function Template:Tmath is complex differentiable at Template:Tmath, but is not complex differentiable anywhere else, especially including in no place close to Template:Tmath (see the Cauchy–Riemann equations, below). So, it is not holomorphic at Template:Tmath.
The relationship between real differentiability and complex differentiability is the following: If a complex function Template:Tmath is holomorphic, then Template:Tmath and Template:Tmath have first partial derivatives with respect to Template:Tmath and Template:Tmath, and satisfy the Cauchy–Riemann equations:[6]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \mbox{and} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\,}
or, equivalently, the Wirtinger derivative of Template:Tmath with respect to Template:Tmath, the complex conjugate of Template:Tmath, is zero:[7]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial\bar{z}} = 0,}
which is to say that, roughly, Template:Tmath is functionally independent from Template:Tmath, the complex conjugate of Template:Tmath.
If continuity is not given, the converse is not necessarily true. A simple converse is that if Template:Tmath and Template:Tmath have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then Template:Tmath is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if Template:Tmath is continuous, Template:Tmath and Template:Tmath have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then Template:Tmath is holomorphic.[8]
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 Template:Tmath is a complex function that is complex differentiable about a point Template:Tmath then (as we did earlier in the article) we can write Template:Tmath and then the complex derivative of the function can be written as Template:Tmath.[9]
Terminology
[edit | edit source]The term holomorphic was introduced in 1875 by Charles Briot and Jean-Claude Bouquet, two of Augustin-Louis Cauchy'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 derived from μέρος (méros) meaning "part". A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane.[10] Cauchy had instead used the term synectic.[11]
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
[edit | edit source]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.[12] That is, if functions Template:Tmath and Template:Tmath are holomorphic in a domain Template:Tmath, then so are Template:Tmath, Template:Tmath, Template:Tmath, and Template:Tmath. Furthermore, Template:Tmath is holomorphic if Template:Tmath has no zeros in Template:Tmath; otherwise it is meromorphic.
If one identifies Template:Tmath with the real plane Template:Tmath, 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 equations.[6]
Every holomorphic function can be separated into its real and imaginary parts Template:Tmath, and each of these is a harmonic function on Template:Tmath (each satisfies Laplace's equation Template:Tmath), with Template:Tmath the harmonic conjugate of Template:Tmath.[13] Conversely, every harmonic function Template:Tmath on a simply connected domain Template:Tmath is the real part of a holomorphic function: If Template:Tmath is the harmonic conjugate of Template:Tmath, unique up to a constant, then Template:Tmath is holomorphic.
Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes:[14]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_\gamma f(z)\,\mathrm{d}z = 0.}
Here Template:Tmath is a rectifiable path in a simply connected complex domain Template:Tmath whose start point is equal to its end point, and Template:Tmath is a holomorphic function.
Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary.[14] Furthermore: Suppose Template:Tmath is a complex domain, Template:Tmath is a holomorphic function and the closed disk Template:Tmath is completely contained in Template:Tmath. Let Template:Tmath be the circle forming the boundary of Template:Tmath. Then for every Template:Tmath in the interior of Template:Tmath:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z}
where the contour integral is taken counter-clockwise.
The derivative Template:Tmath can be written as a contour integral[14] using Cauchy's differentiation formula:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,}
for any simple loop positively winding once around Template:Tmath, and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},}
for infinitesimal positive loops Template:Tmath around Template:Tmath.
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.[15]
Every holomorphic function is analytic. That is, a holomorphic function Template:Tmath has derivatives of every order at each point Template:Tmath in its domain, and it coincides with its own Taylor series at Template:Tmath in a neighbourhood of Template:Tmath. In fact, Template:Tmath coincides with its Taylor series at Template:Tmath 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 Template:Tmath is an integral domain if and only if the open set Template:Tmath is connected.[7] In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.
From a geometric perspective, a function Template:Tmath is holomorphic at Template:Tmath if and only if its exterior derivative Template:Tmath in a neighbourhood Template:Tmath of Template:Tmath is equal to Template:Tmath for some continuous function Template:Tmath. It follows from
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z}
that Template:Tmath is also proportional to Template:Tmath, implying that the derivative Template:Tmath is itself holomorphic and thus that Template:Tmath is infinitely differentiable. Similarly, Template:Tmath implies that any function Template:Tmath that is holomorphic on the simply connected region Template:Tmath is also integrable on Template:Tmath.
For a path Template:Tmath from Template:Tmath to Template:Tmath lying entirely in Template:Tmath, define
In light of the Jordan curve theorem and the generalized Stokes' theorem, Template:Tmath is independent of the particular choice of path Template:Tmath, and thus Template:Tmath is a well-defined function on Template:Tmath having Template:Tmath, or equivalently Template:Tmath.
Examples
[edit | edit source]All polynomial functions in Template:Tmath with complex coefficients are entire functions (holomorphic in the whole complex plane Template:Tmath), and so are the exponential function Template:Tmath and the trigonometric functions Template:Tmath and Template:Tmath (cf. Euler's formula). The principal branch of the complex logarithm function Template:Tmath is holomorphic on the domain Template:Tmath. The square root function can be defined as Template:Tmath and is therefore holomorphic wherever the logarithm Template:Tmath is. The reciprocal function Template:Tmath is holomorphic on Template:Tmath. (The reciprocal function, and any other rational function, is meromorphic on Template:Tmath.)
As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant. Therefore, the absolute value Template:Tmath, the argument Template:Tmath, the real part Template:Tmath and the imaginary part Template:Tmath are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate Template:Tmath. (The complex conjugate is antiholomorphic.)
Several variables
[edit | edit source]The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function Template:Tmath in Template:Tmath complex variables is analytic at a point Template:Tmath if there exists a neighbourhood of Template:Tmath in which Template:Tmath is equal to a convergent power series in Template:Tmath complex variables;[16] the function Template:Tmath is holomorphic in an open subset Template:Tmath of Template:Tmath if it is analytic at each point in Template:Tmath. Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function Template:Tmath, this is equivalent to Template:Tmath being holomorphic in each variable separately (meaning that if any Template:Tmath coordinates are fixed, then the restriction of Template:Tmath is a holomorphic function of the remaining coordinate). The much deeper Hartogs' theorem proves that the continuity assumption is unnecessary: Template:Tmath 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 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 domains, 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 Template:Tmath-form]] Template:Tmath is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero: Template:Tmath.
Extension to functional analysis
[edit | edit source]The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. For instance, the 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
[edit | edit source]References
[edit | edit source]- ↑ "Analytic functions of one complex variable". Encyclopedia of Mathematics. European Mathematical Society / Springer. 2015 – via encyclopediaofmath.org.
- ↑ Template:SpringerEOM
- ↑ Ahlfors, L. (1979), Complex Analysis (3rd ed.), McGraw-Hill
- ↑ Henrici, P. (1986) [1974, 1977]. Applied and Computational Complex Analysis. Wiley. Three volumes, publ.: 1974, 1977, 1986.
- ↑ Ebenfelt, Peter; Hungerbühler, Norbert; Kohn, Joseph J.; Mok, Ngaiming; Straube, Emil J. (2011). Complex Analysis. Science & Business Media. Springer. ISBN 978-3-0346-0009-5 – via Google.
- ↑ 6.0 6.1 Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall. [In three volumes.]
- ↑ 7.0 7.1 Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601 – via Google.
- ↑ Gray, J.D.; Morris, S.A. (April 1978). "When is a function that satisfies the Cauchy–Riemann equations analytic?". The American Mathematical Monthly. 85 (4): 246–256. doi:10.2307/2321164. JSTOR 2321164.
- ↑ Ponce Campuzano, Juan Carlos (14 August 2021). "2.3: Complex Differentiation". Complex Analysis – A Visual and Interactive Introduction. LibreTexts. Retrieved 15 June 2025.
- ↑ The original French terms were holomorphe and méromorphe.
Briot, Charles Auguste; Bouquet, Jean-Claude (1875). "§15 fonctions holomorphes". Théorie des fonctions elliptiques (2nd ed.). Gauthier-Villars. pp. 14–15.
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.
[When a function is continuous, 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 functions which enjoy these properties in the full extent of the plane. [...] ¶ A rational fraction admits as poles the 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.] Harkness, James; Morley, Frank (1893). "5. Integration". A Treatise on the Theory of Functions. Macmillan. p. 161. - ↑ Briot & Bouquet had previously also adopted Cauchy’s term synectic (synectique in French), in the 1859 first edition of their book. Briot, Charles Auguste; Bouquet, Jean-Claude (1859). "§10". Théorie des fonctions doublement périodiques. Mallet-Bachelier. p. 11.
- ↑ Henrici, Peter (1993) [1986]. Applied and Computational Complex Analysis. Wiley Classics Library. 3 (Reprint ed.). New York – Chichester – Brisbane – Toronto – Singapore: John Wiley & Sons. ISBN 0-471-58986-1. MR 0822470. Zbl 1107.30300 – via Google.
- ↑ Evans, L.C. (1998). Partial Differential Equations. American Mathematical Society.
- ↑ 14.0 14.1 14.2 Lang, Serge (2003). Complex Analysis. Springer Verlag GTM. Springer Verlag.
- ↑ Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). New York: McGraw-Hill Book Co. ISBN 978-0-07-054234-1. MR 0924157.
- ↑ Gunning and Rossi. Analytic Functions of Several Complex Variables. p. 2.
Further reading
[edit | edit source]- Blakey, Joseph (1958). University Mathematics (2nd ed.). London, UK: Blackie and Sons. OCLC 2370110.