Definition:Harmonic

Disambiguation

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Harmonic may refer to:

• Algebra:
  • Harmonic mean: the reciprocal of the arithmetic mean of the reciprocals

• Number theory:
  • Harmonic numbers: $\ds H_n = \sum_{k \mathop = 1}^n \frac 1 k$
  • General harmonic numbers: $\ds H_n^{\paren r} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$
  • Harmonic integer or Ore number: a positive integer whose harmonic mean of its divisors is an integer
  • Leibniz harmonic triangle: a triangular array whose elements are derived from the reciprocals of the elements of Pascal's triangle

• Calculus:
  • Harmonic function: a twice continuously differentiable function $f: U \to \R$ (where $U$ is an open set of $\R^n$) which satisfies Laplace's equation

• Analysis:
  • Harmonic analysis: the study of functions by expressing them as the sum of series of a family of functions such as sines and cosines

• Real analysis:
  • Harmonic sequence: a sequence $\sequence {a_k}$ in $\R$ defined as $h_k = \dfrac 1 {a + k d}$
  • Harmonic series: the series defined as $\ds \sum_{n \mathop = 1}^\infty \frac 1 n = 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \cdots$
  • Alternating harmonic series: the series defined as $1 - \dfrac 1 2 + \dfrac 1 3 - \dfrac 1 4 + \dotsb$, also known as Mercator's constant
  • General harmonic series: a series defined as $\ds \sum_{n \mathop = 1}^\infty x_n$ where $\sequence {\size {x_n} }$ is a harmonic sequence
  • Harmonic progression: a harmonic sequence or a harmonic series

• Laplace's Equation:
  • Harmonic: a solution $\phi$ to Laplace's equation in $2$ dimensions
  • Spherical harmonic: a solution $\phi$ to Laplace's equation in $3$ dimensions, expressed in spherical coordinates
  • Surface harmonic: a spherical harmonic whose radial coordinate equals $1$
    • Tesseral harmonic: a surface harmonic of the form $\cos m \phi \, \map { {P_n}^m} {\cos \theta}$ or $\sin m \phi \, \map { {P_n}^m} {\cos \theta}$ such that $m < n$
    • Sectoral harmonic: a surface harmonic of the form $\cos m \phi \, \map { {P_n}^m} {\cos \theta}$ or $\sin m \phi \, \map { {P_n}^m} {\cos \theta}$ such that $m = n$
  • Zonal harmonic: the function $\map {P_n} {\cos \theta}$

• Physics:
  • Harmonic potential energy: the potential energy of a physical particle of the form $\map U x = \frac 1 2 k x^2$
  • Harmonic oscillator: a physical particle whose potential energy is that of the harmonic potential
  • Simple harmonic motion: a physical system $S$ whose motion can be expressed in the form $x = A \map \sin {\omega t + \phi}$

• Analytic geometry:
  • Harmonic ratio: a cross-ratio $\set {A, B; C, D}$ of points on a straight line of $A$, $B$, $C$ and $D$ such that $\set {A, B; C, D} = -1$
  • Harmonic range: line segments $AB$ and $PQ$ on a straight line such that $\dfrac {AP} {PB} = -\dfrac {AQ} {QB}$
  • Harmonic conjugates: the points $P$ and $Q$ with respect to $A$ and $B$ where $AB$ and $PQ$ are a harmonic range
  • Harmonic pencil: Lines from a point to a harmonic range

Also defined as

Some sources use the term harmonic as an adjective meaning able to be expressed using sines and cosines.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): harmonic