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Categories: Complex analysis | Potential theory | Theorems

Harnack's principle

In complex analysis, Harnack's principle is a theorem about the behavior of sequences of harmonic functions.

If the functions $u 1 (z)$ , $u 2 (z)$ , ... are harmonic in an open subset $G$ of the complex plane C, and

$u_1(z) \le u_2(z) \le ...$

in every point of $G$ , then the limit

$\lim_{n\to\infty}u_n(z)$

either is infinite in every point of the domain $G$ or it is finite in every point of the domain, in both cases uniformly in each closed subset of $G$ . In the latter case, the function

$u(z) = \lim_{n\to\infty}u_n(z)$

is harmonic in the set $G$ .

Categories: Complex analysis | Potential theory | Theorems

Last updated: 05-29-2005 06:37:27

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10-26-2009 08:16:03
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Harnack's principle