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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

 (d^2x)/(dt^2)+omega^2x=0,
(1)

which has solution

 x=Acos(omegat+phi_1)+Bsin(omegat+phi_2).
(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 f(x,y) which satisfies Laplace's equation

 del ^2f(x,y)=0,
(3)

where del ^2 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,

 del ^2f(x,y)+k^2f(x,y)=0.
(4)

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

 (AC)/(CB)=-(AD)/(DB).
(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 x_i, the harmonic mean H is defined by

 1/H=1/nsum_(i=1)^n1/(x_i).
(6)

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


See also

Harmonic Addition Theorem, Harmonic Conjugate, Harmonic Conjugate Function, Harmonic Coordinates, Harmonic Decomposition, Harmonic Divisor Number, Harmonic Form, Harmonic Function, Harmonic-Geometric Mean, Harmonic Homology, Harmonic Logarithm, Harmonic Map, Harmonic Mean, Harmonic Mean Index, Harmonic Number, Harmonic Range, Harmonic Series, Simple Harmonic Motion

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Cite this as:

Weisstein, Eric W. "Harmonic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Harmonic.html

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