Quasiperiodic function

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In mathematics, a function f is said to be quasiperiodic with quasiperiod (sometimes simply called the period) ω if for certain constants a and b, f satisfies the functional equation

$f(z + \omega) = \exp(az+b) f(z). \$

An example of this is the Jacobi theta function, where

$\vartheta(z+\tau;\tau) = \exp(-2 \pi i z - \pi i \tau)\vartheta(z;\tau),$

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

$f(z + \omega) = f(z)+az+b \$

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

$\zeta(z + \omega) = \zeta(z) + \eta \$

for a fixed constant η when ω is a period of the corresponding Weierstrass ℘ function.

In the special case where $f(z + \omega)=f(z) \$ we say f is periodic with period ω.

[edit] 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.