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...

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...

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}...

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

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

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

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

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

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

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...

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...

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

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

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...

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 + ⋯...

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...

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...

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

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

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

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...

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

triad. This chord has a dominant function. Unlike the dominant triad or dominant seventh, the leading-tone triad functions as a prolongational chord rather...

of this kind of analysis in various ways. For instance, if u is a harmonic function, then the above sort of contradiction does not directly occur, since...

opera. But it seems already to have been an established, if infrequent, harmonic practice by the end of the 17th century, used by Giacomo Carissimi, Arcangelo...

 133–140. Sheldon Axler, Paul Bourdon, Wade Ramey: Bounded Harmonic Functions. In: Harmonic Function Theory (= Graduate Texts in Mathematics 137). Springer...

which deals with the convergence of sequences of harmonic functions. Given a sequence of harmonic functions u1, u2, ... on an open connected subset G of the...

density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of...