Quasiperiodic function

In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function f is quasiperiodic with quasiperiod \omega if f(z + \omega) = g(z,f(z)), where g is a "simpler" function than f. What it means to be "simpler" is vague.

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

:f(z + \omega) = f(z) + C

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

:f(z + \omega) = C f(z)

An example of this is the Jacobi theta function, where

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

shows that for fixed \tau it has quasiperiod \tau; 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. 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 by Floquet theory.

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, \Lambda) = \zeta(z , \Lambda) + \eta (\omega , \Lambda) \

for a z-independent 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 in the period lattice \Lambda.