Quasiperiodic function
Class of functions behaving "like" periodic functions / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Quasiperiodic function?
Summarize this article for a 10 year old
In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function.[1] A function is quasiperiodic with quasiperiod if , where is a "simpler" function than . What it means to be "simpler" is vague.
This article needs additional citations for verification. (January 2023) |
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
- ;\tau )=e^{-2\pi iz-\pi i\tau }\vartheta (z;\tau ),}
shows that for fixed it has quasiperiod ; 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 z-independent η when ω is a period of the corresponding Weierstrass ℘ function.
In the special case where we say f is periodic with period ω in the period lattice .