MorerasTheorem 2002-08-23 18:28:42 2005-05-16 15:11:44 Theorem % 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 \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} Morera's theorem provides the converse of Cauchy's integral theorem. {\bf Theorem} \cite{rudin_real} Suppose $G$ is a region in $\sC$, and $f:G\to \sC$ is a continuous function. If for every closed triangle $\Delta$ in $G$, we have $$\int_{\partial \Delta} f\, dz = 0,$$ then $f$ is analytic on $G$. (Here, $\partial \Delta$ is the piecewise linear \PMlinkname{boundary}{BoundaryInTopology} of $\Delta$.) In particular, if for every rectifiable closed curve $\Gamma$ in $G$, we have $\int_{\Gamma} f\, dz = 0,$ then $f$ is analytic on $G$. Proofs of this can be found most undergraduate books on complex analysis \cite{kreyszig93, silverman}. \begin{thebibliography}{9} \bibitem{rudin_real} W. Rudin, \emph{Real and complex analysis}, 3rd ed., McGraw-Hill Inc., 1987. \bibitem {kreyszig93} E. Kreyszig, \emph{Advanced Engineering Mathematics}, John Wiley \& Sons, 1993, 7th ed. \bibitem{silverman} R.A. Silverman, \emph{Introductory Complex Analysis}, Dover Publications, 1972. \end{thebibliography}