# Bôcher's theorem

Bôcher's theorem In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher.

Contents 1 Bôcher's theorem in complex analysis 2 Bôcher's theorem for harmonic functions 3 See also 4 External links Bôcher's theorem in complex analysis In complex analysis, the theorem states that the finite zeros of the derivative {displaystyle r'(z)} of a non-constant rational function {displaystyle r(z)} that are not multiple zeros are also the positions of equilibrium in the field of force due to particles of positive mass at the zeros of {displaystyle r(z)} and particles of negative mass at the poles of {displaystyle r(z)} , with masses numerically equal to the respective multiplicities, where each particle repels with a force equal to the mass times the inverse distance.

Furthermore, if C1 and C2 are two disjoint circular regions which contain respectively all the zeros and all the poles of {displaystyle r(z)} , then C1 and C2 also contain all the critical points of {displaystyle r(z)} .

Bôcher's theorem for harmonic functions In the theory of harmonic functions, Bôcher's theorem states that a positive harmonic function in a punctured domain (an open domain minus one point in the interior) is a linear combination of a harmonic function in the unpunctured domain with a scaled fundamental solution for the Laplacian in that domain.

See also Marden's theorem External links Marden, Morris (1951-05-01). "Book Review: The location of critical points of analytic and harmonic functions". Bulletin of the American Mathematical Society. 57 (3): 194–205. doi:10.1090/s0002-9904-1951-09490-2. MR 1565303. (Review of Joseph L. Walsh's book.) This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

Categories: Theorems in complex analysisHarmonic functionsMathematical analysis stubs

Si quieres conocer otros artículos parecidos a Bôcher's theorem puedes visitar la categoría Harmonic functions.

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