Editing Analytic function (section)

==Properties of analytic functions==
* The sums, products, and [[function composition|compositions]] of analytic functions are analytic.
* The [[Multiplicative inverse|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 function|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]] <math>\Omega \subseteq \mathbb{C}</math>, the set ''A''(Ω) of all  analytic functions <math>u:\Omega \to \mathbb{C}</math> 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 <math>A_\infty(\Omega)</math> of all [[bounded function|bounded]] analytic functions with the [[supremum norm]] is a [[Banach space]].

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]] inside its [[domain of a function|domain]], then ƒ is zero everywhere on the [[connected space|connected component]] containing the accumulation point. In other words, if (''r<sub>n</sub>'') is a [[sequence]] of distinct numbers such that ƒ(''r''<sub>''n''</sub>)&nbsp;=&nbsp;0 for all ''n'' and this sequence [[limit of a 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 [[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 (physics and chemistry)|degrees of freedom]] than polynomials, they are still quite rigid.