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| In [[mathematics]], a [[function (mathematics)|function]] is said to be '''quasiperiodic''' when it has some similarity to a periodic function but does not meet the strict definition. To be more precise, this means a function <math>f</math> is quasiperiodic with quasiperiod <math>\omega</math> if <math>f(z + \omega) = g(z,f(z))</math>, where <math>g</math> is a "simpler function" than <math>f</math>. Note that what it means to be a simpler function is vague.
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| A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:
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| :<math> f(z + \omega) = f(z) + C </math>
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| Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:
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| :<math> f(z + \omega) = C f(z) </math>
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| A useful example is the function:
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| :<math> f(z) = \sin(Az) + \sin(Bz) </math>
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| If the ratio ''A''/''B'' is rational, this will have a true period, but if ''A''/''B'' is irrational there is no true period, but a succession of increasingly accurate "almost" periods.
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| An example of this is the [[theta function|Jacobi theta function]], where
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| :<math>\vartheta(z+\tau;\tau) = e^{-2\pi iz - \pi i\tau}\vartheta(z;\tau),</math>
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| shows that for fixed τ it has quasiperiod τ; it also is periodic with period one. Another example is provided by the [[Weierstrass sigma function]], which is quasiperiodic in two independent quasiperiods, the periods of the corresponding [[Weierstrass elliptic functions|Weierstrass ℘ function]].
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| Functions with an additive functional equation
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| :<math> f(z + \omega) = f(z)+az+b \ </math>
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| are also called quasiperiodic. An example of this is the [[Weierstrass zeta function]], where
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| :<math> \zeta(z + \omega) = \zeta(z) + \eta \ </math>
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| for a fixed constant η when ω is a period of the corresponding Weierstrass ℘ function.
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| In the special case where <math> f(z + \omega)=f(z) \ </math> we say ''f'' is [[periodic function|periodic]] with period ω.
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| ==Quasiperiodic signals==
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| Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions; instead they have the nature of [[almost periodic function]]s and that article should be consulted. The more vague and general notion of [[quasiperiodicity]] has even less to do with quasiperiodic functions in the mathematical sense.
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| == See also ==
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| * [[Quasiperiodicity]]
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| * [[Quasiperiodic motion]]
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| * [[Almost periodic function]]
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| ==External links==
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| *[http://planetmath.org/encyclopedia/QuasiperiodicFunction.html Quasiperiodic function] at [[PlanetMath]]
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| [[Category:Complex analysis]]
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| [[Category:Types of functions]]
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