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Inmathematics, aquasiperiodic function is afunction that has a certain similarity to aperiodic function.^[1] A function${\displaystyle f}$ is quasiperiodic with quasiperiod${\displaystyle \omega }$ if${\displaystyle f(z+\omega )=g(z,f(z))}$, where${\displaystyle g}$ is a "simpler" function than${\displaystyle f}$. What it means to be "simpler" is vague.

A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:

${\displaystyle f(z+\omega )=f(z)+C}$

Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:

${\displaystyle f(z+\omega )=Cf(z)}$

An example of this is theJacobi theta function, where

${\displaystyle \vartheta (z+\tau ;\tau )=e^{-2\pi iz-\pi i\tau }\vartheta (z;\tau ),}$

shows that for fixed${\displaystyle \tau }$ it has quasiperiod${\displaystyle \tau }$; it also is periodic with period one. Another example is provided by theWeierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the correspondingWeierstrass℘ function.Bloch's theorem says that the eigenfunctions of a periodic Schrödinger equation (or other periodic linear equations) can be found in quasiperiodic form, and a related form of quasi-periodic solution for periodic linear differential equations is expressed byFloquet theory.

Functions with an additive functional equation

${\displaystyle f(z+\omega )=f(z)+az+b}$

are also called quasiperiodic. An example of this is theWeierstrass zeta function, where

${\displaystyle \zeta (z+\omega ,\mathrm{\Lambda })=\zeta (z,\mathrm{\Lambda })+\eta (\omega ,\mathrm{\Lambda })}$

for az-independent η when ω is a period of the corresponding Weierstrass ℘ function.

In the special case where${\displaystyle f(z+\omega )=f(z)}$ we sayf isperiodic with period ω in the period lattice${\displaystyle \mathrm{\Lambda }}$.

Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions in the sense defined here; instead they have the nature ofalmost periodic functions and that article should be consulted. The more vague and general notion ofquasiperiodicity has even less to do with quasiperiodic functions in the mathematical sense.

A useful example is the function:

${\displaystyle f(z)=\sin (Az)+\sin (Bz)}$

If the ratioA/B isrational, this will have a true period, but ifA/B isirrational there is no true period, but a succession of increasingly accurate "almost" periods.

• Quasiperiodic motion

• Quasiperiodic function atPlanetMath

• Complex analysis
• Types of functions

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