### Context

#### Analysis

#### Analytic geometry

# Analytic functions

## Idea

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

## Definitions

Let $V$ and $W$ be complete Hausdorff topological vector spaces, let $W$ be locally convex, let $c$ be an element of $V$, and let $(a_0,a_1,a_2,\ldots)$ be an infinite sequence of homogeneous operator?s from $V$ to $W$ with each $a_k$ of degree $k$.

Given an element $c$ of $V$, consider the infinite series

$\sum_k a_k(x - c)^k$

(a power series). Let $U$ be the interior of the set of $x$ such that this series converges in $W$; we call $U$ the **domain of convergence** of the power series. This series defines a function from $U$ to $W$; we are really interested in the case where $U$ is inhabited, in which case it is a balanced neighbourhood? of $c$ in $V$ (which is Proposition 5.3 of Bochnak–Siciak).

Let $D$ be any subset of $V$ and $f$ any continuous function from $D$ to $W$. This function $f$ is **analytic** if, for every $c \in D$, there is a power series as above with inhabited domain of convergence $U$ such that

$f(x) = \sum_k a_k(x - c)^k$

for every $x$ in both $D$ and $U$. (That $f$ is continuous follows automatically in many cases, including of course the finite-dimensional case.)

## Generalisation

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

###### Theorem

A function from $D \subseteq \mathbb{C}$ to $W$ is differentiable if and only if it is analytic.

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

## References

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).