An analytic function is a function that is locally given by a converging power series.
Let and be complete Hausdorff topological vector spaces, let be locally convex, let be an element of , and let be an infinite sequence of homogeneous? operators from to with each of degree .
Given an element of , consider the infinite series
\sum_k a_k(x - c)
(a power series). Let be the interior of the set of such that this series converges in ; we call the domain of convergence of the power series. This series defines a function from to ; we are really interested in the case where is inhabited, in which case it is a balanced neighbourhood? of in (which is Proposition 5.3 of Bochnak–Siciak).
Let be any subset of and any continuous function from to . This function is analytic if, for every , there is a power series as above with inhabited domain of convergence such that
f(x) = \sum_k a_k(x - c)
for every in both and . (That is continuous follows automatically in many cases, including of course the finite-dimensional case.)
The vector spaces and may be generalised to analytic manifolds and (more generally) analytic spaces. However, these are manifolds and varieties modelled on vector spaces using analytic transition functions, so the notion of analytic function between vector spaces is most fundamental.
Complex-analytic functions of one variable
If is a vector space over the complex numbers, then we have this very nice theorem, due essentially to Édouard Goursat:
A function from to is analytic if and only if it is differentiable.
(Differentiability here is in the usual sense, that the difference quotient converges in .) See holomorphic function and Goursat theorem.
The theory of analytic function was constructed to some extent by
and in full generality by
Textbook accounts include
- Jacek Bochnak and Józef Siciak, Analytic functions in topological vector spaces; Studia Mathematica 39 (1971); (pdf).