Bocher's Theorem
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Bocher's Theorem

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.