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'Morera's theorem'
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| Title of object: |
Morera's theorem |
| Canonical Name: |
MorerasTheorem |
| Type: |
Theorem |
| Created on: |
2002-08-23 18:28:42.843976-04 |
| Modified on: |
2002-08-23 18:28:42.843976-04 |
| Classification: |
msc:30D20 |
Revision comment (for changes between this and next version):
| Changes for correction #1812 ('quantifier'). |
Preamble:
% 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 |
Content:
Let $\Omega$ be an open set in the complex numbers $\mathbb{C}$ and let $f$ be a continuous function on $\Omega$. If for any closed rectangle $R\subset\Omega$, we have
$$\int_{\partial R} f\, dz = 0$$
then $f$ is holomorphic on $\Omega$. |
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