nLab analytic function

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

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

(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

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

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.

Related concepts

• analytic geometry

• holomorphic function, meromorphic function

• smooth function

• analytic (∞,1)-functor

• HoTT book real numbers

References

The theory of analytic function was constructed to some extent by

• M. Krasner (1940)

and in full generality by

• John Tate (1961)

Textbook accounts:

• Robert C. Gunning, Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs (1965)

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

• Stephen Schanuel, Continuous extrapolation to triangular matrices characterizes smooth functions, J. Pure App. Alg. 24, Issue 1 (1982), 59–71. web