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Revision difference : Harnack's principle |
| Version 4 |
Version 3 |
| If the functions \,$u_1(z)$, $u_2(z)$, ... are harmonic in the domain \,$G \subseteq\mathbb{C}$\, and |
If the functions \,$u_1(z)$, $u_2(z)$, ... are harmonic in the domain \,$G \subseteq\mathbb{C}$\, and |
| $$u_1(z) \leqq u_2(z) \leqq ...$$ |
$$u_1(z) \leqq u_2(z) \leqq ...$$ |
| 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 $G$. |
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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