and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R , {\displaystyle f\colon U\to \mathbb...

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving...

In music, function (also referred to as harmonic function) is a term used to denote the relationship of a chord or a scale degree to a tonal centre. Two...

.} As a first consequence of the definition, they are both harmonic real-valued functions on Ω {\displaystyle \Omega } . Moreover, the conjugate of u...

In mathematics, a function f {\displaystyle f} is weakly harmonic in a domain D {\displaystyle D} if ∫ D f Δ g = 0 {\displaystyle \int _{D}f\,\Delta g=0}...

In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure...

zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function: 143 ...

the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions. Informally, the...

mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when...

Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency...

continuously differentiable solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics...

Sometimes such a function is referred to as n-harmonic function, where n ≥ 2 is the dimension of the complex domain where the function is defined. However...

real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps, namely...

sweeping") is a method devised by Henri Poincaré for reconstructing an harmonic function in a domain from its values on the boundary of the domain. In modern...

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually...

mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical...

semitone. Because of this construction, the 7th degree of the harmonic minor scale functions as a leading tone to the tonic because it is a semitone lower...

In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ n = 1 ∞ 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯...

defined on an open subset U of M, is harmonic if each individual coordinate function xi is a harmonic function on U. That is, one requires that Δ g x...

potential theory, a subharmonic function f satisfies Δf ≥ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the boundary...

(Ramsey theory) In mathematics, Radó's theorem is a result about harmonic functions, named after Tibor Radó. Informally, it says that any "nice looking"...

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional...

harmonic function on the boundary of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also inside...

estimate Harmonic maps Harmonic morphisms Holomorphic separability Meromorphic function Quadrature domains Wirtinger derivatives "Analytic functions of one...

potential The class of functions known as harmonic functions, which are the topic of study in potential theory The potential function of a potential game...

include "harmonic" include: Projective harmonic conjugate Cross-ratio Harmonic analysis Harmonic conjugate Harmonic form Harmonic function Harmonic mean Harmonic...

u}{\partial \nu }}\,\mathrm {d} \sigma ',} where: u is the unique harmonic function defined on the region D between Σ and S with the boundary conditions...

is a universal moduli space for the Poisson integral, expressing a harmonic function on a group in terms of its boundary values. A model for the Furstenberg...

effects created by distinct pitches or tones coinciding with one another; harmonic objects such as chords, textures and tonalities are identified, defined...

mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions R 3 → C {\displaystyle...