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Encyclopedia > Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions. 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 signal processing, the analytic signal, or analytic representation, of a signal is defined by: where is the Hilbert transform of and (aka ) is the imaginary unit. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about functions in mathematics. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Series expansion redirects here. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

Contents

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 In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

 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, ... 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. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ...


Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x0 in its domain In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... Series expansion redirects here. ...

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

converges to f(x) for x close enough to x0.


The definition of a complex analytic function is obtained by replacing, in the definitions 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). Typical examples of analytic functions are: In mathematics, several functions are important enough to deserve their own name. ...

  • Any polynomial (real or complex) is an analytic function. This is because if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion will vanish, and so this series will be trivially convergent.
  • The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x0 (as in the definition) but for all values of x (real or complex).

Typical examples of functions that are not analytic are: In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... The exponential function is one of the most important functions in mathematics. ... Sine redirects here. ... -1... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

  • The absolute value function when defined on the set of real numbers or complex numbers is not analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulas in different regions) are in general not analytic.
  • The complex conjugate function, is not complex analytic, although its restriction to the real line is real analytic.

In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

Properties of analytic functions

  • The sums, products, and compositions of analytic functions are analytic.
  • The reciprocal of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose derivative is nowhere zero. (See also the Lagrange inversion theorem.)
  • Any analytic function is smooth, that is, infinitely differentiable. The converse is not true; in fact, in a certain sense, the analytic functions are rather sparse compared to the infinitely differentiable functions.
  • For any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a Banach space with respect to the supremum norm. The fact that uniform limits of analytic functions are analytic is an easy consequence of Morera's theorem.

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, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... The reciprocal function: y = 1/x. ... For other uses, see Derivative (disambiguation). ... In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ... In complex analysis, Moreras theorem states that if the integral of a continuous complex-valued function f of a complex variable along every simple closed curve within an open set is zero, that is, if for C any simple closed curve, then f is differentiable at every point in... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... In mathematics, the domain of a function is the set of all input values to the function. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...


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. For other senses of this word, see sequence (disambiguation). ... 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. Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...


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 fact there are many such functions, and the space of real analytic functions is a proper subspace of the space of smooth functions. In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of 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, the term analytic function is synonymous with holomorphic function. In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and is analytic at a if in some open disk centered at a it can be expanded as... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...


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. This statement is clearly false for real analytic functions, as illustrated by Liouvilles theorem in complex analysis states that every bounded (i. ...

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. 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 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 spaces in general. ... In geometry, a disk is the region in a plane contained inside of a circle. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...


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). 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. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...


See also

In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

References

John B. Conway is a mathematician at University of Tennessee. ... Graduate Texts in Mathematics (GTM) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. ...

External links

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

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Analytic functions can be thought of as a bridge between polynomials and general functions.
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