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

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.)