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'Harnack's principle'
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| Title of object: |
Harnack's principle |
| Canonical Name: |
HarnacksPrinciple |
| Type: |
Theorem |
| Created on: |
2005-01-22 10:02:39 |
| Modified on: |
2005-01-22 10:10:00 |
| Classification: |
msc:31A05, msc:31B05, msc:31C05, msc:30F15 |
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:
If the functions \,$u_1(z)$, $u_2(z)$, ... are harmonic in the domain \,$G \subseteq\mathbb{C}$\, and
$$u_1(z) \leqq u_2(z) \leqq ...$$
in every point of $G$, then \,$\lim_{n\to\infty}u_n(z)$\, either is infinite in every point of the domain or it is finite in every point of the domain, in both cases uniformly in each \PMlinkname{closed}{ClosedSet} subdomain of $G$. \,In the latter case, the function \,$u(z) = \lim_{n\to\infty}u_n(z)$\, is harmonic in the domain $G$. |
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