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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an analytic function is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that is locally given by a
convergent Convergent is an adjective for things that wikt:converge, converge. It is commonly used in mathematics and may refer to: *Convergent boundary, a type of plate tectonic boundary * Convergent (continued fraction) * Convergent evolution * Convergent s ...
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
. There exist both real analytic functions and complex analytic functions. Functions of each type are
infinitely differentiable is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...

infinitely differentiable
, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
about ''x''0 converges to the function in some
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
for every ''x''0 in its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
.


Definitions

Formally, a function f is ''real analytic'' on an
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
D in the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
is
convergent Convergent is an adjective for things that wikt:converge, converge. It is commonly used in mathematics and may refer to: *Convergent boundary, a type of plate tectonic boundary * Convergent (continued fraction) * Convergent evolution * Convergent s ...
to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an
infinitely differentiable function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...

infinitely differentiable function
such that the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
at any point x_0 in its domain : T(x) = \sum_^ \frac (x-x_0)^ converges to f(x) for x in a neighborhood of x_0 pointwise. The set of all real analytic functions on a given set D is often denoted by C^(D). A function f defined on some subset of the real line is said to be real analytic at a point x if there is a neighborhood D of x on which f is real analytic. The definition of a ''complex analytic function'' is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.


Examples

Typical examples of analytic functions are: * All
elementary function In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...
s: ** All
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s: if a polynomial has degree ''n'', any terms of degree larger than ''n'' in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin (1 ...
. ** The
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

exponential function
is analytic. Any Taylor series for this function converges not only for ''x'' close enough to ''x''0 (as in the definition) but for all values of ''x'' (real or complex). ** The
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

trigonometric function
s,
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

logarithm
, and the power functions are analytic on any open set of their domain. * Most
special function Special functions are particular mathematical function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), ...
s (at least in some range of the complex plane): **
hypergeometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s **
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematici ...
s **
gamma function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

gamma function
s Typical examples of functions that are not analytic are: * The
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet. * The
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
function ''z'' → ''z''* is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from \mathbb^ to \mathbb^. * Other
non-analytic smooth function
non-analytic smooth function
s, and in particular any smooth function f with compact support, i.e. f \in C^\infty_0(\R^n), cannot be analytic on \R^n.


Alternative characterizations

The following conditions are equivalent: 1. f is real analytic on an open set D. 2. There is a complex analytic extension of f to an open set G \subset \mathbb which contains D. 3. f is real smooth and for every
compact set In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...
K \subset D there exists a constant C such that for every x \in K and every non-negative integer k the following bound holds :: \left , \frac(x) \right , \leq C^ k! Complex analytic functions are exactly equivalent to
holomorphic functions Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
, and are thus much more easily characterized. For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the
Fourier–Bros–Iagolnitzer transform In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization. Let U \subset \R^n be an open set, and let f: U \to \R. Then f is real analytic on U if and only if f \in C^\infty(U) and for every compact K \subseteq U there exists a constant C such that for every multi-index \alpha \in \Z_^n the following bound holds : \sup_ \left , \frac(x) \right , \leq C^\alpha!


Properties of analytic functions

* The sums, products, and compositions of analytic functions are analytic. * The
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...

reciprocal
of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
is nowhere zero. (See also the
Lagrange inversion theorem In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (numbe ...
.) * Any analytic function is
smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...

smooth
, that is, infinitely differentiable. The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. For the complex numbers, the converse does hold, and in fact any function differentiable ''once'' on an open set is analytic on that set (see "analyticity and differentiability" below). * For any
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
Ω ⊆ C, the set ''A''(Ω) of all analytic functions ''u'' : Ω → C is a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are Complete space, complete with ...
with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of
Morera's theorem In complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigat ...
. The set \scriptstyle A_\infty(\Omega) of all bounded analytic functions with the
supremum norm frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded function Image:Bounded and ...
is a
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function ƒ has an
accumulation point In mathematics, a limit point (or cluster point or accumulation point) of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood ...
inside its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
, then ƒ is zero everywhere on the connected component containing the accumulation point. In other words, if (''rn'') is a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
of distinct numbers such that ƒ(''r''''n'') = 0 for all ''n'' and this sequence converges to a point ''r'' in the domain of ''D'', then ƒ is identically zero on the connected component of ''D'' containing ''r''. This is known as the Principle of Permanence. Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component. These statements imply that while analytic functions do have more
degrees of freedom Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...
than polynomials, they are still quite rigid.


Analyticity and differentiability

As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or ''C''). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see
non-analytic smooth function
non-analytic smooth function
. In fact there are many such functions. The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that any complex function differentiable (in the complex sense) in an open set is analytic. Consequently, in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
, the term ''analytic function'' is synonymous with ''
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
''.


Real versus complex analytic functions

Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts. According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by :f(x)=\frac. Also, if a complex analytic function is defined in an open
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
around a point ''x''0, its power series expansion at ''x''0 is convergent in the whole open ball (
holomorphic functions are analytic In complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigat ...
). This statement for real analytic functions (with open ball meaning an open interval of the real line rather than an open
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
of the complex plane) is not true in general; the function of the example above gives an example for ''x''0 = 0 and a ball of radius exceeding 1, since the power series diverges for , ''x'',  ≥ 1. Any real analytic function on some
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function ƒ(''x'') defined in the paragraph above is a counterexample, as it is not defined for ''x'' = ±''i''. This explains why the Taylor series of ƒ(''x'') diverges for , ''x'',  > 1, i.e., the
radius of convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is 1 because the complexified function has a
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.


Analytic functions of several variables

One can define analytic functions in several variables by means of power series in those variables (see
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in 2 or more complex dimensions: * Zero sets of complex analytic functions in more than one variable are never
discrete Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...
. This can be proved by
Hartogs's extension theoremIn mathematics, precisely in the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the Singularity (mathematics), singularities of holomorphic functions of several variables. Informally, it states that ...
. * Domains of holomorphy for single-valued functions consist of arbitrary (connected) open sets. In several complex variables, however, only some connected open sets are domains of holomorphy. The characterization of domains of holomorphy leads to the notion of pseudoconvexity.


See also

*
Cauchy–Riemann equations In the field of complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis ...
*
Holomorphic function In mathematics, a holomorphic function is a complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), ...
* Paley–Wiener theorem * Quasi-analytic function *
Infinite compositions of analytic functionsIn mathematics, infinite compositions of analytic function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...


Notes


References

* *


External links

* *
Solver for all zeros of a complex analytic function that lie within a rectangular region by Ivan B. Ivanov
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