quasiperiodic function

A functionMathworldPlanetmath f is said to have a quasiperiod p if there exists a function g such that


In the special case where g is identically equal to 1, we call f a periodic functionMathworldPlanetmath, and we say that p is a period of f or that f has periodicity p.

Except for the special case of periodicity noted above, the notion of quasiperiodicity is somewhat loose and fuzzy. Strictly speaking, many functions could be regarded as quasiperiodic if one defines g(z)=f(z+p)/f(z). In order for the term “quasiperiodic” not to be trivial, it is customary to reserve its use for the case where the function g is, in some vague, intuitive sense, simpler than the function f. For instance, no one would call the function f(z)=z2+1 quasiperiodic even though it meets the criterion of the definition if we set g(z)=(z2+2z+2)/(z2+1) because the rational function g is “more complicated” than the polynomialPlanetmathPlanetmath f. On the other hand, for the gamma functionDlmfDlmfMathworldPlanetmath, one would say that 1 is a quasiperiod because Γ(z+1)=zΓ(z) and the function g(z)=z is a “much simpler” function than the gamma function.

Note that the every complex numberMathworldPlanetmathPlanetmath can be said to be a quasiperiod of the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath. The term “quasiperiod” is most frequently used in connection with theta functionsDlmfMathworld.

Also note that almost periodic functions are quite a different affair than quasiperiodic functions — there, one is dealing with a precise notion.

Title quasiperiodic function
Canonical name QuasiperiodicFunction
Date of creation 2013-03-22 14:40:38
Last modified on 2013-03-22 14:40:38
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 13
Author rspuzio (6075)
Entry type Definition
Classification msc 30A99
Related topic ComplexTangentAndCotangent
Related topic CounterperiodicFunction
Defines quasiperiod
Defines quasiperiodicity
Defines period
Defines periodic function
Defines periodic
Defines periodicity