Holomorphic function

File:Non-holomorphic complex conjugate.svg

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 limitAhlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979).

:$f\text{'}(z\_0)\; =\; \backslash lim\_\{z\; \backslash to\; z\_0\}\; \backslash frac\{f(z)\; -\; f(z\_0)\}\{\; z\; -\; z\_0\; \}.$

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 {{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 real differentiability: It is linear and obeys the product rule, quotient rule, and chain rule.{{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.

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 of {{tmath|z_0}}.

{{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 }}

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 $\backslash textstyle\; f(z)\; =\; |z|\backslash vphantom\{l\}^2\; =\; z\backslash bar\{z\}$ 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:

{{cite book

|last=Markushevich |first=A.I.

|year=1965

|title=Theory of Functions of a Complex Variable

|publisher=Prentice-Hall

}} [In three volumes.]

:$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; =\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; y\}\; \backslash qquad\; \backslash mbox\{and\}\; \backslash qquad\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; =\; -\backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}\backslash ,$

or, equivalently, the Wirtinger derivative of {{tmath|f}} with respect to {{tmath|\bar z}}, the complex conjugate of {{tmath|z}}, is zero:

{{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

}}

:$\backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash bar\{z\}\}\; =\; 0,$

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.

{{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

}}

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.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, 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.}} {{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}} Cauchy had instead used the term synectic.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 }}

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.

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.

{{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

}}

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 {{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 equations.

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}}.

{{cite book

|first=L.C. |last=Evans |author-link=Lawrence C. Evans

|year=1998

|title=Partial Differential Equations

|publisher=American Mathematical Society

}}

Conversely, every harmonic function {{tmath|u(x, y)}} on a 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 vanishes:

{{cite book

|first = Serge |last = Lang | author-link = Serge Lang

| year = 2003

| title = Complex Analysis

| series = Springer Verlag GTM

| publisher = Springer Verlag

}}

:$\backslash oint\_\backslash gamma\; f(z)\backslash ,\backslash mathrm\{d\}z\; =\; 0.$

Here {{tmath|\gamma}} is a rectifiable path in a simply connected 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 is completely determined by its values on the disk's boundary. Furthermore: Suppose {{tmath|U \subset \C}} is a complex domain, {{tmath|f\colon U \to \C}} is a holomorphic function and the closed disk $D\; \backslash equiv\; \backslash \{\; z\; :\; |\; z\; -\; z\_0\; |\; \backslash leq\; r\; \backslash \}$ is completely contained in {{tmath|U}}. Let {{tmath|\gamma}} be the circle forming the boundary of {{tmath|D}}. Then for every {{tmath|a}} in the interior of {{tmath|D}}:

:$f(a)\; =\; \backslash frac\{\; 1\; \}\{2\backslash pi\; i\}\; \backslash oint\_\backslash gamma\; \backslash frac\{f(z)\}\{z-a\}\backslash ,\backslash mathrm\{d\}z$

where the contour integral is taken counter-clockwise.

The derivative {{tmath|{f'}(a)}} can be written as a contour integral using Cauchy's differentiation formula:

:$f\text{'}\backslash !(a)\; =\; \backslash frac\{\; 1\; \}\{2\backslash pi\; i\}\; \backslash oint\_\backslash gamma\; \backslash frac\{f(z)\}\{(z-a)^2\}\backslash ,\backslash mathrm\{d\}z,$

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

:$f\text{'}\backslash !(a)\; =\; \backslash lim\backslash limits\_\{\backslash gamma\backslash to\; a\}\; \backslash frac\{\; i\; \}\{2\backslash mathcal\{A\}(\backslash gamma)\}\; \backslash oint\_\{\backslash gamma\}f(z)\backslash ,\backslash mathrm\{d\}\backslash bar\{z\},$

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

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.

{{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

}}

Every 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. 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 {{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

:$0\; =\; \backslash mathrm\{d\}^2\; f\; =\; \backslash mathrm\{d\}(f\text{'}\backslash ,\backslash mathrm\{d\}z)\; =\; \backslash mathrm\{d\}f\text{'}\; \backslash wedge\; \backslash mathrm\{d\}z$

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 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} }}.)

All polynomial functions in {{tmath|z}} with complex coefficients are entire functions (holomorphic in the whole complex plane {{tmath|\C}}), and so are the 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 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 reciprocal function {{tmath|\tfrac{1}{z} }} is holomorphic on {{tmath| \C \smallsetminus \{ 0 \} }}. (The reciprocal function, and any other rational function, is meromorphic on {{tmath|\C}}.)

As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant. Therefore, the absolute value {{nobr|$|z|$,}} the argument {{tmath|\arg z}}, the real part {{tmath|\operatorname{Re}(z)}} and the 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.)

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;

{{cite book

|last1=Gunning |last2=Rossi |name-list-style=and

|title=Analytic Functions of Several Complex Variables

|page=2

}}

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 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 {{tmath|\alpha}} is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero: {{tmath|1= \bar{\partial}\alpha = 0}}.

{{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 or Gateaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers.

{{div col begin|colwidth=18em}}

• Antiderivative (complex analysis)
• Antiholomorphic function
• Biholomorphy
• Cauchy's estimate
• Harmonic maps
• Harmonic morphisms
• Holomorphic separability
• Meromorphic function
• Quadrature domains
• Wirtinger derivatives

{{div col end}}

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• {{cite book

|last=Blakey |first=Joseph

|year=1958

|title=University Mathematics |edition=2nd

|publisher=Blackie and Sons

|location=London, UK

|oclc=2370110

}}

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

{{Authority control}}

Category:Analytic functions