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Not to be confused with Analytic signal. In Mathematics and Signal processing, the analytic representation of a real-valued function or signal facilitates many mathematical manipulations of the signal

In mathematics, an analytic function is a function that is locally given by a convergent power series. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + Analytic functions can be thought of as a bridge between polynomials and general functions. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if it is equal to its Taylor series in some neighborhood. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

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Definitions

Formally, a function f is real analytic on an open set D in the real line if for any x0 in D one can write

f(x) = \sum_{n=0}^\infty a_n \left( x-x_0 \right)^n = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots

in which the coefficients a0, a1, . In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, the real numbers may be described informally in several different ways . . are real numbers and the series is convergent for x in a neighborhood of x0. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with

Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x0 in its domain

T(x) = \sum_{n=0}^{\infin} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n}

is convergent for x close enough to x0 and its value equals f(x). In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

The definition of a complex analytic function is obtained by replacing everywhere above "real" with "complex", and "real line" with "complex plane".

Examples

Most special functions are analytic (at least in some range of the complex plane). Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis Typical examples of analytic functions are:

Typical examples of functions that are not analytic are:

Properties of analytic functions

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 f has an accumulation point inside its domain, then f is zero everywhere on the connected component containing the accumulation point. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of

More formally this can be stated as follows. If (rn) is a sequence of distinct numbers such that f(rn) = 0 for all n and this sequence converges to a point r in the domain of D, then f is identically zero on the connected component of D containing r. In Mathematics, a sequence is an ordered list of objects (or events The limit of a sequence is one of the oldest concepts in Mathematical analysis.

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 than polynomials, they are still quite rigid. For information on degrees of freedom in other sciences see Degrees of freedom.

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 which are not analytic: see the following example. In Mathematics, Smooth functions (also called infinitely differentiable functions and Analytic functions are two very important types of functions. In fact there are many such functions, and the space of real analytic functions is a proper subspace of the space of smooth 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. In Complex analysis, a field of Mathematics, a complex -valued function f of a complex variable z is holomorphic Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

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. In Complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded Entire function must be constant This statement is clearly false for real analytic functions, as illustrated by

f(x)=\frac{1}{x^2+1}.

Also, if a complex analytic function is defined in an open ball around a point x0, its power series expansion at x0 is convergent in the whole ball. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric This is not true in general for real analytic functions. (Note that an open ball in the complex plane would be a disk, while on the real line it would be an interval. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set )

Any real analytic function on some open set 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 f (x) defined in the paragraph above is a counterexample, as it is not defined for x = ±i.

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 f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + 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 when working in 2 or more dimensions. For instance, zero sets of complex analytic functions in more than one variables are never discrete.

See also

References

External links

Dictionary

analytic function

-noun

  1. (analysis) a real valued function which is uniquely defined through its derivatives at one point
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