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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.<ref> Template:Cite encyclopedia </ref>
Holomorphic functions are also sometimes referred to as regular functions.<ref>Template:SpringerEOM</ref> 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.
DefinitionEdit
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<ref>Ahlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979).</ref>
- <math>f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{ z - z_0 }.</math>
This is the same definition as for the derivative of a real function, except that all quantities are complex. In particular, the limit is taken as the complex number 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.<ref>Template:Cite book Three volumes, publ.: 1974, 1977, 1986.</ref>
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.<ref> Template:Cite book </ref> 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 <math>\textstyle f(z) = |z|\vphantom{l}^2 = z\bar{z}</math> is complex differentiable at Template:Tmath, but is not complex differentiable anywhere else, esp. 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:<ref name=Mark> Template:Cite book [In three volumes.] </ref>
- <math>\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \mbox{and} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\,</math>
or, equivalently, the Wirtinger derivative of Template:Tmath with respect to Template:Tmath, the complex conjugate of Template:Tmath, is zero:<ref name=Gunning> Template:Cite book </ref>
- <math>\frac{\partial f}{\partial\bar{z}} = 0,</math>
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.<ref> Template:Cite journal </ref>
TerminologyEdit
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.<ref>The original French terms were holomorphe and méromorphe. Template:Pb Template:Cite book Template:Pb Template:Cite book</ref> Cauchy had instead used the term synectic.<ref>Briot & Bouquet had previously also adopted Cauchy’s term synectic (synectique in French), in the 1859 first edition of their book. Template:Pb Template:Cite book</ref>
Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.
PropertiesEdit
Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> Template:Cite book </ref> 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.<ref name=Mark/>
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.<ref> Template:Cite book </ref> 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:<ref name=Lang> Template:Cite book </ref>
- <math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math>
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.<ref name=Lang/> Furthermore: Suppose Template:Tmath is a complex domain, Template:Tmath is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \leq r \} </math> 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:
- <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 counter-clockwise.
The derivative Template:Tmath can be written as a contour integral<ref name=Lang /> using Cauchy's differentiation formula:
- <math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math>
for any simple loop positively winding once around Template:Tmath, and
- <math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math>
for infinitesimal positive loops 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.<ref> Template:Cite book </ref>
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. <ref name=Gunning/> 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
- <math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math>
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 Template:Tmath; 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 Template:Tmath.)
ExamplesEdit
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:Nobr 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 variablesEdit
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;<ref> Template:Cite book </ref> 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 analysisEdit
Template:Main article 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 alsoEdit
- Antiderivative (complex analysis)
- Antiholomorphic function
- Biholomorphy
- Cauchy's estimate
- Harmonic maps
- Harmonic morphisms
- Holomorphic separability
- Meromorphic function
- Quadrature domains
- Wirtinger derivatives