Harmonic

The word "harmonic" has several distinct meanings in mathematics, none of which is obviously related to the others. Simple harmonic motion or "harmonic oscillation" refers to oscillations with a sinusoidal waveform. Such functions satisfy the differential equation

(1)

which has solution

(2)

The word harmonic analysis is therefore used to describe Fourier series, which breaks an arbitrary function into a superposition of sinusoids.

In complex analysis, a harmonic function refers to a real-valued function which satisfies Laplace's equation

(3)

where is the Laplacian. Although this definition is similar to that of harmonic oscillation, it omits the second term in the differential equation. The Helmholtz differential equation is obtained if it is added back in,

(4)

For distances along a line segment, a harmonic range is a set of four collinear points , , , and arranged such that

(5)

This use of the term probably arises from the use of "harmonics" to refer to ratios of notes in small integers producing an attractive sound, known in music theory as "harmony."

For a set of data points , the harmonic mean is defined by

(6)

The connection of this use of "harmonic" with the preceding ones is not obvious.

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