Quasiperiodic Function

Quasiperiodic Function

In mathematics, a function is said to be quasiperiodic when it has some similarity to a periodic function but does not meet the strict definition. To be more precise, this means a function is quasiperiodic with quasiperiod if, where is a "simpler function" than . Note that what it means to be a simpler function is vague.

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:

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.

An example of this is the Jacobi theta function, where

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

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

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

In the special case where we say f is periodic with period ω.