30A99-IdentityTheoremOfHolomorphicFunctions.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{IdentityTheoremOfHolomorphicFunctions}
\pmcreated{2013-03-22 16:47:05}
\pmmodified{2013-03-22 16:47:05}
\pmowner{rspuzio}{6075}
\pmmodifier{rspuzio}{6075}
\pmtitle{identity theorem of holomorphic functions}
\pmrecord{11}{39016}
\pmprivacy{1}
\pmauthor{rspuzio}{6075}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{30A99}
\pmsynonym{rigidity theorem for analytic functions}{IdentityTheoremOfHolomorphicFunctions}
\pmrelated{IdentityTheoremOfPowerSeries}
\pmrelated{IdentityTheorem}

\endmetadata

% this is the default PlanetMath preamble.  as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
 \usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}

\begin{document}
If the functions $f$ and $g$ are holomorphic in a domain $D$ of 
the complex plane and the equation
\begin{align}
f(z) = g(z)
\end{align}   
is true in an infinite subset $S$ of $D$ having an accumulation point $z_0$ in $D$, then (1) is true in the whole $D$.

\textbf{Remark.}\, The subset $S$ may be e.g. some neighbourhood of $z_0$ or some arc containing $z_0$.
%%%%%
%%%%%
\end{document}