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Functions in mathematics
and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb
Harmonic_function
Special mathematical functions defined on the surface of a sphere
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving
Spherical_harmonics
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
Positive_harmonic_function
Area of mathematical analysis
Harmonic analysis is an area of mathematical analysis that emerged from the study of harmonic functions, and especially their boundary behavior. The methods
Harmonic_analysis
Musical term
In music, function (also harmonic function or tonal function) denotes the relationship of a chord or scale degree to a tonal centre. Two main theories
Function_(music)
Second-order partial differential equation
continuously differentiable solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics
Laplace's_equation
Concept in mathematics
the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions. Informally, the
Harmonic_map
Solutions of Lamé's equation
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It
Lamé_function
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and
Harmonic_number
In Euclidean space, a measure of that set's "size"
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
Capacity_of_a_set
In mathematics, Radó's theorem is a result about harmonic functions, named after Tibor Radó. Informally, it says that any "nice looking" shape without
Radó's theorem (harmonic functions)
Radó's_theorem_(harmonic_functions)
mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical
Harmonic_measure
Harmonic functions as solutions to Laplace's equation
mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" dates from 19th-century physics when
Potential_theory
Concept in mathematics
.} As a first consequence of the definition, they are both harmonic real-valued functions on Ω {\displaystyle \Omega } . Moreover, the conjugate of u
Harmonic_conjugate
Method for reconstructing a harmonic function in a domain
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
Balayage
Differential operator in mathematics
density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of
Laplace_operator
Model in probability theory
subharmonic function f {\displaystyle f} satisfies Δ f ≥ 0 {\displaystyle \Delta f\geq 0} . Any subharmonic function bounded above by a harmonic function for
Martingale (probability theory)
Martingale_(probability_theory)
Class of mathematical functions
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
Subharmonic_function
Mathematical terminology
include "harmonic" include: Projective harmonic conjugate Cross-ratio Harmonic analysis Harmonic conjugate Harmonic form Harmonic function Harmonic mean Harmonic
Harmonic_(mathematics)
Musical scale
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
Harmonic_minor_scale
Aspect of music
effects created by distinct pitches or tones coinciding with one another; harmonic objects such as chords, textures and tonalities are identified, defined
Harmony
Complex-differentiable (mathematical) function
estimate Harmonic maps Harmonic morphisms Holomorphic separability Meromorphic function Quadrature domains Wirtinger derivatives "Analytic functions of one
Holomorphic_function
Divergent sum of positive unit fractions
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ i = 1 ∞ 1 i = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 +
Harmonic_series_(mathematics)
Type of musical chord
triad. This chord has a dominant function. Unlike the dominant triad or dominant seventh, the leading-tone triad functions as a prolongational chord rather
Diminished_triad
Quantum mechanical model
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually
Quantum_harmonic_oscillator
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}
Weakly_harmonic_function
Polynomial whose Laplacian is zero
Harmonic function Spherical harmonics Zonal spherical harmonics Multilinear polynomial Walsh, J. L. (1927). "On the Expansion of Harmonic Functions in
Harmonic_polynomial
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
Pluriharmonic_function
Inequality for Harmonic Functions
Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). Harnack's inequality
Harnack's_inequality
Major chord in music theory
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
Neapolitan_chord
Physical system that responds to a restoring force proportional to displacement
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional
Harmonic_oscillator
Theorem on the convergence of harmonic functions
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
Harnack's_principle
Use of Roman Numeral symbols in the musical analysis of chords
is a type of harmonic analysis in which chords are represented by Roman numerals, which encode the chord's degree and harmonic function within a given
Roman_numeral_analysis
Cellular automaton
⌋ {\displaystyle \lfloor .\rfloor } the floor function. For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the
Abelian_sandpile_model
Succession of musical chords
In a musical composition, a chord progression or harmonic progression (informally chord changes, used as a plural, or simply changes) is a succession of
Chord_progression
Certain vector fields are the sum of an irrotational and a solenoidal vector field
133–140. Sheldon Axler, Paul Bourdon, Wade Ramey "Bounded Harmonic FunctionsHarmonic Function Theory (= Graduate Texts in Mathematics 137). Springer, New
Helmholtz_decomposition
potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also
Kelvin_transform
Mathematical concept
certain Möbius transformations. Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane.
Poisson_kernel
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
Harmonic_coordinates
Inverse of the average of the inverses of a set of numbers
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is sometimes used for ratios and rates such as speeds, and is
Harmonic_mean
Three notes in intervals of a third
Harvard University Press, 1950): 704, s.v. Spacing. Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of its
Triad_(music)
Method of solution to differential equations
well-known property of harmonic functions, that if the value or normal derivative is known on a bounding surface, then the value of the function inside the volume
Green's_function
{\displaystyle r(z)} . In the theory of harmonic functions, Bôcher's theorem states that a positive harmonic function in a punctured domain (an open domain
Bôcher's_theorem
of related results in the mathematical study of the regularity of harmonic functions on sufficiently smooth domains by Oliver Dimon Kellogg. In the first
Kellogg's_theorem
real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps, namely
Harmonic_morphism
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Type of differential equation
solid is a harmonic function. It is usually a matter of straightforward computation to check whether or not a given function is harmonic. For instance
Partial_differential_equation
Fourth-order PDE in continuum mechanics
harmonic functions and v ( x , y ) {\displaystyle v(x,y)} is a harmonic conjugate of u ( x , y ) {\displaystyle u(x,y)} . Just as harmonic functions in
Biharmonic_equation
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
and often corresponding to a form of Green's function. Four methods of constructing the harmonic function are widely employed: the Perron method; the Schwarz
Uniformization_theorem
Branch of mathematics
spaces, measure spaces, and function spaces. Its major areas include complex analysis, functional analysis, measure theory, harmonic analysis, and the theory
Mathematical_analysis
Integral transform and linear operator
{y}{\pi \,\left(x^{2}+y^{2}\right)}}} Furthermore, there is a unique harmonic function v defined in the upper half-plane such that F(z) = u(z) + i v(z) is
Hilbert_transform
Mathematical form
In potential theory (the study of harmonic functions) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar
Dirichlet_form
Type of problem involving ODEs or PDEs
value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's
Boundary_value_problem
Mathematical measure space associated to a random walk
semisimple Lie group. The Poisson formula states that given a positive harmonic function f {\displaystyle f} on the unit disc D = { z ∈ C : | z | < 1 } {\displaystyle
Poisson_boundary
Mathematical technique
In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar
Perron_method
Green's function for Laplacian
for Isaac Newton, who first discovered it and proved that it was a harmonic function in the special case of three variables, where it served as the fundamental
Newtonian_potential
Mathematical theorem in complex analysis
{\displaystyle \ln |f(z)|} is a harmonic function. Since z 0 {\displaystyle z_{0}} is a local maximum for this function also, it follows from the maximum
Maximum_modulus_principle
Mathematical formula in complex analysis
|f(re^{i\theta })|\,d\theta ,} which is the mean-value property of the harmonic function log | f ( z ) | {\displaystyle \log |f(z)|} . An equivalent statement
Jensen's_formula
Mathematical series
the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion
Multipole_expansion
Concept in potential theory
solution to Poisson's equation. Dirichlet's principle states that, if the function u ( x ) {\displaystyle u(x)} is the solution to Poisson's equation Δ u
Dirichlet's_principle
Mathematics principle in complex analysis
apply to harmonic functions. Kelvin transform Method of image charges Schwarz function Cartan, Henri. Elementary theory of analytic functions of one or
Schwarz_reflection_principle
Simultaneous use of multiple musical keys
same time. Polyvalence or polyvalency is the use of more than one harmonic function, from the same key, at the same time. Some examples of bitonality
Polytonality
Theorem in complex analysis
maximum principle if they achieve their maxima at the boundary of D. Harmonic functions and, more generally, solutions of elliptic partial differential equations
Maximum_principle
The harmonic function U. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued harmonic function U on
Planar_Riemann_surface
Chord that contains the interval of an augmented sixth
Baroque to the Romantic periods, augmented sixth chords had the same harmonic function: as a chromatically altered predominant chord (typically, an alteration
Augmented_sixth_chord
real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point
Hopf_lemma
Analytic function with prescribed zeros
also a consequence of the maximum principle for harmonic functions, applied to the harmonic function log ( | f ( z ) | ) {\displaystyle \log(|f(z)|)}
Blaschke_product
Velocity field as the gradient of a scalar function
the help of the harmonic function φ {\displaystyle \varphi } and its conjugate harmonic function ψ {\displaystyle \psi } (stream function), incompressible
Potential_flow
Typically linear operator defined in terms of differentiation of functions
an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers
Differential_operator
Musical term
towards resolution of the dominant. The predominant harmonic function is part of the fundamental harmonic progression of many classical works. The submediant
Predominant_chord
Theoretical description of Earth's gravimetric shape
differential equation (6) (the Laplace equation) are called spherical harmonic functions. They take the forms: where spherical coordinates (r, θ, φ) are used
Geopotential spherical harmonic model
Geopotential_spherical_harmonic_model
Western music history period (c. 1650 to 1900)
a union between harmonic function and counterpoint. In other words, individual melodic lines, when taken together, express harmonic unity and goal-oriented
Common_practice_period
Tonal degree of the diatonic scale
dominant function. Leading-tone triads and leading-tone seventh chords may also have dominant function. In very much conventionally tonal music, harmonic analysis
Dominant_(music)
Wave with frequency an integer multiple of the fundamental frequency
1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also
Harmonic
exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational
List_of_real_analysis_topics
analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle). For a rational PR function, the number
Positive-real_function
American mathematician (born 1949)
Bourdon, and Wade Ramey) Harmonic Function Theory, second edition, Graduate Texts in Mathematics, Springer, 2001. Harmonic Function Theory software, a Mathematica
Sheldon_Axler
German engineer and mathematician
who contributed to the field of special functions and complex analysis being a pioneer in the field of harmonic analysis. Schuster was born in Munich,
Edmund_Schuster
To-and-fro periodic motion in science and engineering
In mechanics and physics, simple harmonic motion (sometimes abbreviated as SHM) is a special type of periodic motion an object experiences by means of
Simple_harmonic_motion
Solutions of the Laplace equation in spherical polar coordinates
mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions R 3 → C {\displaystyle
Solid_harmonics
Formula in differential geometry
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold ( M , g ) {\displaystyle (M,g)} to the Ricci curvature
Bochner's_formula
Topics referred to by the same term
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
Potential_function
Characteristic property of holomorphic functions
That is, u is a harmonic function. This means that the divergence of the gradient is zero, and so the fluid is incompressible. The function v also satisfies
Cauchy–Riemann_equations
Extension of the scalar spherical harmonics for use with vector fields
harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions
Vector_spherical_harmonics
Alteration of the original shape of a signal
harmonic distortion plus noise (THD+N). In telecommunications and signal processing, a noise-free system can be characterised by a transfer function,
Distortion
Wave shaped like the sine function
waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is simple harmonic motion; as rotation, it corresponds
Sine_wave
Eleventh chord used in jazz music
augmented eleventh, including maj7♯11, add9♯11, and 6♯11. Lydian chords may function as subdominants or substitutes for the tonic in major keys. The compound
Lydian_chord
Russian mathematician (born 1989)
граничных свойствах гармонических функций, On boundary properties of harmonic functions). He works at the Chebyshev Mathematics Laboratory of the Saint Petersburg
Aleksandr Logunov (mathematician)
Aleksandr_Logunov_(mathematician)
Metric in relativity
y)(dt+dz)^{2}-dx^{2}-dy^{2}-dz^{2}} for any arbitrary function f. If f is a harmonic function with respect to x and y, then the corresponding Peres metric
Peres_metric
Theorem in complex analysis
adapted to the case where the harmonic function f {\displaystyle f} is merely bounded above or below. See Harmonic function#Liouville's theorem. Another
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Surface that locally minimizes its area
boundary. This definition ties minimal surfaces to harmonic functions and potential theory. Harmonic definition: If X = ( x 1 , x 2 , x 3 ) : M → R 3 {\displaystyle
Minimal_surface
{\displaystyle L^{2}} norm of an harmonic function on a given sphere in terms of the L 2 {\displaystyle L^{2}} norm of this function on two spheres, one with
Three_spheres_inequality
in this case, a Bäcklund transformation of a harmonic function is just a conjugate harmonic function. The above properties mean, more precisely, that
Bäcklund_transform
Functions such that f(–x) equals f(x) or –f(x)
no even harmonics. If the function f(x) is even, a cosine input will produce no odd harmonics (but may contain a DC component). If the function is neither
Even_and_odd_functions
Elliptic partial differential equation
equation Uniqueness theorem for Poisson's equation Weak formulation Harmonic function Heat equation Potential theory Jackson, Julia A.; Mehl, James P.;
Poisson's_equation
Musical theories of Hugo Riemann
and by a concept of harmonic functions. Riemann's "dualist" system for relating triads was adapted from earlier 19th-century harmonic theorists. The term
Riemannian_theory
Real-valued function
In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation
Bounded_mean_oscillation
Calculation of complex statistical distributions
the use of bounded harmonic functions. Definition (Harmonic function) A measurable function h {\displaystyle h} is said to be harmonic for the chain ( X
Markov_chain_Monte_Carlo
Topics referred to by the same term
chemical valence Polyvalence (music), the musical use of more than one harmonic function of a tonality simultaneously Polyvalent antibody, a group of antibodies
Polyvalence
Numbers in the Roman numeral system
are often numbered using Roman numerals. In Roman numeral analysis, harmonic function is identified using Roman numerals. Individual strings of stringed
Roman_numerals
HARMONIC FUNCTION
HARMONIC FUNCTION
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Boy/Male
French American Hebrew
Girl/Female
Greek Latin
Daughter of Ares.
Girl/Female
American, Australian, British, Christian, English, French, Greek, Latin
A State of Order or Agreement; Unity; Concord; Harmony; Agreement
Girl/Female
American, Australian, British, Chinese, Christian, English, French, Greek, Latin
A State of Order or Agreement; A Beautiful Blending; Agreement; Concord; Musical Combination of Chords; Harmony; Joining
Boy/Male
Welsh
Harmony.
Boy/Male
Muslim
Harmony
Female
English
English name derived from the vocabulary word harmony, from Greek Harmonia, HARMONY means "concord, harmony."
Male
English
English surname transferred to forename use, from the German personal name Harman, HARMON means "bold/hardy man."
Female
Greek
(ΑÏμονία) Greek name HARMONIA means "concord, harmony." In mythology, this is the name of the daughter of Ares and Aphrodite. Her Latin name is Concordia.
Girl/Female
Christian & English(British/American/Australian)
Harmony
Girl/Female
American, British, English, Greek, Latin
A State of Order or Agreement; Unity; Concord; Musically in Tune; A Tuneful Sound
Girl/Female
Latin American
Concord.
Girl/Female
Latin
Harmony.
Female
English
Variant spelling of English Harmony, HARMONIE means "concord, harmony."
Boy/Male
American, Australian, British, Chinese, Christian, English, French, German, Greek, Hebrew
Man of the Army; Army Man; Noble; Name of a Place During Biblical Period; Hardy Man; Variant of Herman
Boy/Male
Indian
Harmony
Girl/Female
Latin
Harmony.
Surname or Lastname
Irish (mainly County Louth)
Irish (mainly County Louth) : generally of English origin (see 1); but sometimes also used as a variant of Harman or Hardiman, i.e. an Anglicized form of Gaelic Ó hArgadáin (see Hargadon).English : variant spelling of Harman 1.
HARMONIC FUNCTION
HARMONIC FUNCTION
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
The Flute
Boy/Male
Indian
Lord Vishnu, Imperishable, Indestructible, Immovable
Surname or Lastname
English (Dorset)
English (Dorset) : ethnic name for a Cornishman.
Girl/Female
French
Crowned with laurels.
Girl/Female
Arabic
Soul; Group Leader
Boy/Male
Arabic
Fortunate; Prosperous
Girl/Female
Tamil
Noble, Secret, Righteous
Boy/Male
Christian, Gujarati, Hindu, Indian, Kannada, Oriya, Telugu
Warrior
Girl/Female
Hindu, Indian
Good Girl
Male
African
a prince royal of Ethiopia.
HARMONIC FUNCTION
HARMONIC FUNCTION
HARMONIC FUNCTION
HARMONIC FUNCTION
HARMONIC FUNCTION
n.
A musical note produced by a number of vibrations which is a multiple of the number producing some other; an overtone. See Harmonics.
n.
The just adaptation of parts to each other, in any system or combination of things, or in things, or things intended to form a connected whole; such an agreement between the different parts of a design or composition as to produce unity of effect; as, the harmony of the universe.
a.
Not harmonic; inharmonious; discordant; dissonant.
n.
One who understands the principles of harmony or is skillful in applying them in composition; a musical composer.
v. i.
To agree in vocal or musical effect; to form a concord; as, the tones harmonize perfectly.
n.
One who shows the agreement or harmony of corresponding passages of different authors, as of the four evangelists.
n.
One of a religious sect, founded in Wurtemburg in the last century, composed of followers of George Rapp, a weaver. They had all their property in common. In 1803, a portion of this sect settled in Pennsylvania and called the village thus established, Harmony.
n.
A literary work which brings together or arranges systematically parallel passages of historians respecting the same events, and shows their agreement or consistency; as, a harmony of the Gospels.
a.
Alt. of Harmonical
a.
Concordant; musical; consonant; as, harmonic sounds.
n.
Alt. of Harmonite
v. i.
To agree in action, adaptation, or effect on the mind; to agree in sense or purport; as, the parts of a mechanism harmonize.
n.
See Harmonic suture, under Harmonic.
pl.
of Harmony
a.
Relating to harmony, -- as melodic relates to melody; harmonious; esp., relating to the accessory sounds or overtones which accompany the predominant and apparent single tone of any string or sonorous body.
a.
Not harmonic.
n.
Concord or agreement in facts, opinions, manners, interests, etc.; good correspondence; peace and friendship; as, good citizens live in harmony.
v. t.
To accompany with harmony; to provide with parts, as an air, or melody.
a.
Producing mathematically perfect harmony or concord; sweetly or perfectly harmonious.
a.
Of, pertaining to, or obtained from, carbon; as, carbonic oxide.