+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- # Analytic functions * table of contents {: toc} ## Idea An _analytic function_ is a [[function]] that is locally given by a converging [[power series]]. ## Definitions Let $V$ and $W$ be [[complete space|complete Hausdorff]] [[topological vector spaces]], let $W$ be [[locally convex space|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 set|inhabited]], in which case it is a [[balanced neighbourhood]] of $c$ in $V$ (which is Proposition 5.3 of [Bochnak--Siciak](#BS)). 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 manifold]]s and (more generally) [[analytic space]]s. However, these are [[manifolds]] and [[varieties]] modelled on vector spaces using analytic [[transition functions]], so the notion of an analytic function between vector spaces is the 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]]: +-- {: .un_theorem} ###### Theorem A function from $D \subseteq \mathbb{C}$ to $W$ is [[differentiable function|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]] * [[Fabius function]] * [[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: * {#GunningRossi} [[Robert C. Gunning]], [[Hugo Rossi]], *Analytic functions of several complex variables*, Prentice-Hall Inc., Englewood Cliffs (1965) * {#BS} Jacek Bochnak and Józef Siciak, _Analytic functions in topological vector spaces_; Studia Mathematica 39 (1971); ([pdf](http://matwbn.icm.edu.pl/ksiazki/sm/sm39/sm3916.pdf)). * {#Schanuel} [[Stephen Schanuel]], _Continuous extrapolation to triangular matrices characterizes smooth functions_, J. Pure App. Alg. 24, Issue 1 (1982), 59–71. [web](http://www.sciencedirect.com/science/journal/00224049/24/1) [[!redirects analytic function]] [[!redirects analytic functions]] [[!redirects analytic map]] [[!redirects analytic maps]] [[!redirects complex analytic]]