# Definition:Harmonic

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## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**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

- 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**