nLab
analytic function

Analytic functions

Idea

An analytic function is a function that is locally given by a converging power series.

Definitions

Let VV and WW be complete Hausdorff topological vector spaces, let WW be locally convex, let cc be an element of VV, and let (a 0,a 1,a 2,)(a_0,a_1,a_2,\ldots) be an infinite sequence of homogeneous operator?s from VV to WW with each a ka_k of degree kk.

Given an element cc of VV, consider the infinite series

ka k(xc) \sum_k a_k(x - c)

(a power series). Let UU be the interior of the set of xx such that this series converges in WW; we call UU the domain of convergence of the power series. This series defines a function from UU to WW; we are really interested in the case where UU is inhabited, in which case it is a balanced neighbourhood? of cc in VV (which is Proposition 5.3 of Bochnak–Siciak).

Let DD be any subset of VV and ff any continuous function from DD to WW. This function ff is analytic if, for every cDc \in D, there is a power series as above with inhabited domain of convergence UU such that

f(x)= ka k(xc) f(x) = \sum_k a_k(x - c)

for every xx in both DD and UU. (That ff is continuous follows automatically in many cases, including of course the finite-dimensional case.)

Generalisation

The vector spaces VV and WW 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 WW is a vector space over the complex numbers, then we have this very nice theorem, due essentially to Édouard Goursat:

Theorem

A function from DD \subseteq \mathbb{C} to WW is differentiable if and only if it is analytic.

(Differentiability here is in the usual sense, that the difference quotient converges in WW.) See holomorphic function and Goursat theorem.

References

The theory of analytic function was constructed to some extent by

  • M. Krasner (1940)

and in full generality by

Textbook accounts include

  • Jacek Bochnak and Józef Siciak, Analytic functions in topological vector spaces; Studia Mathematica 39 (1971); (pdf).

Revised on December 25, 2013 04:01:21 by Colin Tan (182.55.186.152)