Bôcher's theorem

Bôcher's theorem in complex analysis

In complex analysis, the theorem states that the finite zeros of the derivative r ′ ( z ) of a nonconstant rational function 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 r ( z ) and particles of negative mass at the poles of 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 r ( z ) , then C₁ and C₂ also contain all the critical points of r ( z ) .

Bôcher's theorem for harmonic functions

In the theory of harmonic functions, Bôcher's theorem states that a positive harmonic function in 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.