Definition:Harmonic


 * 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


 * 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 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