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

Maxime Bôcher Maxime Bôcher (August 28, 1867 – September 12, 1918) was an American mathematician who published about 100 papers on differential equations, series, and algebra. He also wrote elementary texts such as ''Trigonometry'' and ''Analytic Geometry'' ...

Bôcher's theorem in complex analysis

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...

, the theorem states that the finite zeros of the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

r'(z)

of a non- constant

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

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 In theoretical physics, negative mass is a type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and show some strange properties such as th ...

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 ''C''₁ and ''C''₂ are two disjoint circular regions which contain respectively all the zeros and all the poles of

r(z)

, then ''C''₁ and ''C''₂ also contain all the critical points of

r(z)

Bôcher's theorem for harmonic functions

In the theory of

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, ...

s, 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 In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...

for the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is ...

in that domain.

External links

* (Review of

Joseph L. Walsh __NOTOC__ Joseph Leonard Walsh (September 21, 1895 – December 6, 1973) was an American mathematician who worked mainly in the field of analysis. The Walsh function and the Walsh–Hadamard code are named after him. The Grace–Walsh–Szeg ...

's book.) Theorems in complex analysis Harmonic functions {{mathanalysis-stub

Bôcher's theorem in complex analysis

Bôcher's theorem for harmonic functions

See also

External links