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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. | |||
A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation: | |||
:<math> f(z + \omega) = f(z) + C </math> | |||
Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation: | |||
:<math> f(z + \omega) = C f(z) </math> | |||
A useful example is the function: | |||
:<math> f(z) = \sin(Az) + \sin(Bz) </math> | |||
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. | |||
An example of this is the [[theta function|Jacobi theta function]], where | |||
:<math>\vartheta(z+\tau;\tau) = e^{-2\pi iz - \pi i\tau}\vartheta(z;\tau),</math> | |||
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]]. | |||
Functions with an additive functional equation | |||
:<math> f(z + \omega) = f(z)+az+b \ </math> | |||
are also called quasiperiodic. An example of this is the [[Weierstrass zeta function]], where | |||
:<math> \zeta(z + \omega) = \zeta(z) + \eta \ </math> | |||
for a fixed constant η when ω is a period of the corresponding Weierstrass ℘ function. | |||
In the special case where <math> f(z + \omega)=f(z) \ </math> we say ''f'' is [[periodic function|periodic]] with period ω. | |||
==Quasiperiodic signals== | |||
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. | |||
== See also == | |||
* [[Quasiperiodicity]] | |||
* [[Quasiperiodic motion]] | |||
* [[Almost periodic function]] | |||
==External links== | |||
*[http://planetmath.org/encyclopedia/QuasiperiodicFunction.html Quasiperiodic function] at [[PlanetMath]] | |||
[[Category:Complex analysis]] | |||
[[Category:Types of functions]] | |||
Revision as of 05:08, 10 December 2013
In mathematics, a 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 is quasiperiodic with quasiperiod if , where is a "simpler function" than . Note that what it means to be a simpler function is vague.
A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:
Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:
A useful example is the function:
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.
An example of this is the Jacobi theta function, where
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 ℘ function.
Functions with an additive functional equation
are also called quasiperiodic. An example of this is the Weierstrass zeta function, where
for a fixed constant η when ω is a period of the corresponding Weierstrass ℘ function.
In the special case where we say f is periodic with period ω.
Quasiperiodic signals
Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions; instead they have the nature of almost periodic functions 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.