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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: |
2006-10-06 15:11:43 |
| Classification: |
msc:31A05, msc:30F15 |
Preamble:
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\usepackage{amssymb}
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%\usepackage{psfrag}
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Content:
If the functions\, $u_1(z)$, $u_2(z)$, \ldots\, are \PMlinkname{harmonic}{HarmonicFunction} in the domain\, $G \subseteq\mathbb{C}$\, and
$$u_1(z) \leqq u_2(z) \leqq \ldots$$
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 \PMlinkname{uniformly}{UniformConvergence} 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 $G$ (cf. limit function of sequence). |
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