Analytic function

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Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at each point f(x) for D is often denoted by 𝒞ω if the domain is understood.

A function D of f.

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 i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.

In complex analysis, a function is called analytic in an open set ⁠n⁠, any terms of degree larger than ⁠x0⁠ (as in the definition) but for all values of (0,∞) or the complement in the Riemann sphere of any simple arc connecting 1

Complex analytic functions are exactly equivalent to holomorphic functions, 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 the multivariable case, real analytic functions satisfy a direct generalization of the third characterization. Let f:U→ℝ⁠. Then f∈C∞(U) and for every compact α∈ℤ≥0n the following bound holds Ω⊆ℂ⁠, the set u:Ω→ℂ is a Fréchet space 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. The set f⁠ has an accumulation point inside its domain, then ⁠f(rn)=0⁠ for all ⁠D⁠, then ⁠r⁠. This is known as the identity theorem.

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.

As noted above, any function (real or complex) is infinitely differentiable on a neighborhood where it is equal to a convergent power series. In a neighborhood of a point of analyticity, there is a convergent power series equal to the function in the neighborhood, so the function is infinitely differentiable there. There exist smooth real functions that are not analytic: see 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. Any complex-differentiable function in an open disc centered at x0⁠, its power series expansion at ⁠1⁠, since the power series ⁠f(x)⁠ defined in the paragraph above is a counterexample, as it is not defined for ⁠|x|>1⁠, i.e., the radius of convergence is ⁠0⁠ and no further poles within the open disc of radius a. More generally, for any power series +∞, such that the power series converges absolutely for a in the real case, or a disc centered at sin⁡z, and z=0 has radius of convergence f(x)=11+x2 is smooth for all real 1, because the corresponding complex function has singularities at α>0 is not an integer, then the binomial series α, the function z=−1 is not a failure of convergence of the power series, nor a pole or essential singularity, but the branching of the analytic continuation. In effect,

Because analytic functions are locally represented by power series, a value of the function on one small neighborhood can sometimes determine values on a larger region. This process is called analytic continuation. Starting with a power series representation in one neighborhood, one can then sometimes use this to define the function on overlapping neighborhoods, and continue this process along paths in the domain.

Analytic continuation is unique when it exists on a connected domain. More precisely, if two analytic functions on a connected open set agree on a nonempty open subset, or more generally on a set having an accumulation point in the domain, then they agree everywhere on that connected open set. This is a form of the identity theorem.

However, analytic continuation need not be possible everywhere, and it need not be single-valued. For example, the natural logarithm is locally analytic on 2πi. For this reason, a single-valued branch of the logarithm can be defined on domains such as f(z)=1−z is analytic, where the branch is such that z=1, but transforms to its negative after a single loop. The Riemann surface associated with an algebraic function is a finite ramified cover of the Riemann sphere.

Many special functions are first defined by a power series or an integral formula on a restricted domain and then extended by analytic continuation. For example, the Riemann zeta function is initially defined by the Dirichlet series s=1.

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 in two or more complex dimensions:

• Zero sets of complex analytic functions in more than one variable are never discrete if they are non-empty. This can be proved by Hartogs's extension theorem.
• 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.

Analogous notions of analyticity can be formulated over other complete valued fields, the real and complex numbers being the two most prominent ones where the absolute value is archimedean. Analytic functions can also be defined over non-Archimedean local fields, such as the p-adic numbers K is a complete valued field, a function on a neighborhood of a point K. In the non-Archimedean case, convergence is governed by an ultrametric absolute value, and the resulting theory differs significantly from both real and complex analysis.

For example, a power series over ℤp={x∈ℚp∣|x|p≤1} if and only if |an|K→0. The reason is the ultrametric criterion: if |∑n=NManxn|K≤maxN≤n≤M|an|.

More generally, on the closed disc |bn||π|mn→0. Here

1. ↑ This implies uniform convergence as well in a (possibly smaller) neighborhood of ⁠
   Strichartz, Robert S. (1994). A guide to distribution theory and Fourier transforms. Boca Raton: CRC Press. ISBN 0-8493-8273-4. OCLC 28890674.
2. ↑ Krantz & Parks 2002, p. 15.
3. ↑ Ahlfors, Lars V. (1979), Complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-000657-7, Chapter 8.
4. ↑ Serre 1979.
5. ↑ Schikhof 1984.
6. ↑ Bosch, Güntzer & Remmert 1984.