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 .
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 (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.Analytic functions of one complex variable, Encyclopedia of Mathematics. (European Mathematical Society ft. Springer, 2015)
Holomorphic functions are also sometimes referred to as regular functions.
A holomorphic function whose domain is the whole complex plane is called an entire function.
The phrase "holomorphic at a point " means not just differentiable at , but differentiable everywhere within some neighbourhood of  in the complex plane.
Definition
thumb|The function  is not complex differentiable at zero, because as shown above, the value of  varies depending on the direction from which zero is approached.
Along the real axis,  equals the function  and the limit is , while along the imaginary axis,  equals  and the limit is .
Other directions yield yet other limits.
Given a complex-valued function  of a single complex variable, the derivative of  at a point  in its domain is defined by the limitAhlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979).
f'(z_0) = \lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }.
This is the same as the definition of the derivative for real functions, except that all of the quantities are complex.
In particular, the limit is taken as the complex number  approaches , and must have the same value for any sequence of complex values for  that approach  on the complex plane.
If the limit exists, we say that  is complex differentiable at the point .
This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule.Henrici, P., Applied and Computational Complex Analysis (Wiley).
[Three volumes: 1974, 1977, 1986.]
If  is complex differentiable at every point  in an open set , we say that  is holomorphic on .
We say that  is holomorphic at the point  if  is complex differentiable on some neighbourhood of .Peter Ebenfelt, Norbert Hungerbühler, Joseph J. Kohn, Ngaiming Mok, Emil J. Straube (2011) Complex Analysis Springer Science & Business Media We say that  is holomorphic on some non-open set  if it is holomorphic in a neighbourhood of .
As a pathological non-example, the function given by  is complex differentiable at exactly one point (), and for this reason, it is not holomorphic at  because there is no open set around  on which  is complex differentiable.
The relationship between real differentiability and complex differentiability is the following: If a complex function  is holomorphic, then  and  have first partial derivatives with respect to  and , and satisfy the Cauchy–Riemann equations:Markushevich, A.I.,Theory of Functions of a Complex Variable (Prentice-Hall, 1965).
[Three volumes.]
\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  with respect to , the complex conjugate of , is zero:
\frac{\partial f}{\partial\overline{z}} = 0,
which is to say that, roughly,  is functionally independent from  the complex conjugate of .
If continuity is not given, the converse is not necessarily true.
A simple converse is that if  and  have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then  is holomorphic.
A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if  is continuous,  and  have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then  is holomorphic..
Terminology
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.
<br>  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.
Today, the term "holomorphic function" is sometimes preferred to "analytic function".
An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions.
The term "analytic" is however also in wide use.
Properties
Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero..
That is, if functions  and  are holomorphic in a domain , then so are , , , and .
Furthermore,  is holomorphic if  has no zeros in , or is meromorphic otherwise.
If one identifies  with the real plane , 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 , and each of these is a harmonic function  on  (each satisfies Laplace's equation ), with  the harmonic conjugate of ..
Conversely, every harmonic function  on a simply connected domain  is the real part of a holomorphic function: If  is the harmonic conjugate of , unique up to a constant, then  is holomorphic.
Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes:
\oint_\gamma f(z)\,dz = 0.
Here  is a rectifiable path in a simply connected complex domain  whose start point is equal to its end point, and  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  is a complex domain,  is a holomorphic function and the closed disk  is completely contained in .
Let  be the circle forming the boundary of .
Then for every  in the interior of :
f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,dz
where the contour integral is taken counter-clockwise.
The derivative  can be written as a contour integral using Cauchy's differentiation formula:
f'(a) = {1 \over 2\pi i} \oint_\gamma {f(z) \over (z-a)^{2}}\,dz,
for any simple loop positively winding once around , and
f'(a) = \lim\limits_{\gamma\to a}\frac i{2\mathcal{A}(\gamma)}\oint_{\gamma}f(z)\,d\bar z,
for infinitesimal positive loops  around .
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.
Every holomorphic function is analytic.
That is, a holomorphic function  has derivatives of every order at each point  in its domain, and it coincides with its own Taylor series at  in a neighbourhood of .
In fact,  coincides with its Taylor series at  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  is an integral domain if and only if the open set  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  is holomorphic at  if and only if its exterior derivative  in a neighbourhood  of  is equal to  for some continuous function .
It follows from
\textstyle 0 = d^2 f = d(f^\prime dz) = df^\prime \wedge dz
that  is also proportional to , implying that the derivative  is itself holomorphic and thus that  is infinitely differentiable.
Similarly,  implies that any function  that is holomorphic on the simply connected region  is also integrable on .
(For a path  from  to  lying entirely in , define  F_\gamma(z) = F_0 + \int_\gamma f\,dz; in light of the Jordan curve theorem and the generalized Stokes' theorem,  is independent of the particular choice of path , and thus  is a well-defined function on  having  and .)
Examples
All polynomial functions in  with complex coefficients are entire functions (holomorphic in the whole complex plane ), and so are the exponential function  and the trigonometric functions \cos{z} = \tfrac12\bigl(\exp(iz) + \exp(-iz)\bigr) and \sin{z} = -\tfrac12i\bigl(\exp(iz) - \exp(-iz)\bigr) (cf. Euler's formula).
The principal branch of the complex logarithm function  is holomorphic on the domain  The square root function can be defined as \sqrt{z} = \exp\bigl(\tfrac12 \log z\bigr) and is therefore holomorphic wherever the logarithm  is.
The reciprocal function  is holomorphic on  (The reciprocal function, and any other rational function, is meromorphic on .)
As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant.
Therefore, the absolute value , the argument , the real part  and the imaginary part  are not holomorphic.
Another typical example of a continuous function which is not holomorphic is the complex conjugate  (The complex conjugate is antiholomorphic.)
Several variables
The definition of a holomorphic function generalizes to several complex variables in a straightforward way.
Let  to be polydisk and also, denote an open subset of , and let .
The function  is analytic at a point  in  if there exists an open neighbourhood of  in which  is equal to a convergent power series in  complex variables.Gunning and Rossi, Analytic Functions of Several Complex Variables, p.
2.  Define  to be holomorphic if it is analytic at each point in its domain.
Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function , this is equivalent to  being holomorphic in each variable separately (meaning that if any  coordinates are fixed, then the restriction of  is a holomorphic function of the remaining coordinate).
The much deeper Hartogs' theorem proves that the continuity hypothesis is unnecessary:  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  is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero, .
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 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
Antiderivative (complex analysis)
Antiholomorphic function
Biholomorphy
Holomorphic separability
Meromorphic function
Quadrature domains
Harmonic maps
Harmonic morphisms
Wirtinger derivatives
References
Further reading
External links
