Quasiperiodic function

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: An example of this is the Jacobi theta function, where shows that for fixed

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.

An example of this is the Weierstrass zeta function, where for a z-independent η when ω is a period of the corresponding Weierstrass ℘ function.

Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions in the sense defined here; 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.

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.