In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, 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. F ...
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 rat ...
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
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 the ...
at the
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
of
, 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
, then ''C''
1 and ''C''
2 also contain all the critical
points of
.
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,
: \f ...
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 ad ...
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 the ...
in that domain.
See also
*
Marden's theorem
In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its deriva ...
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–Szegő ...
's book.)
Theorems in complex analysis
Harmonic functions
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