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Revision difference : Harnack's principle |
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If the functions\, $u_1(z)$, $u_2(z)$, \ldots are \PMlinkname{harmonic}{HarmonicFunction} in the domain\, $G \subseteq\mathbb{C}$\, and
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If the functions \,$u_1(z)$, $u_2(z)$, ... are \PMlinkname{harmonic}{HarmonicFunction} in the domain \,$G \subseteq\mathbb{C}$\, and
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$$u_1(z) \leqq u_2(z) \leqq \ldots$$
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$$u_1(z) \leqq u_2(z) \leqq ...$$
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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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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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