Quasiperiodic function

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Quasiperiodic function" – news · newspapers · books · scholar · JSTOR (January 2023) (Learn how and when to remove this message)

In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic 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 the Jacobi 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 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

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

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

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

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

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