# quasiperiodic function

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

 $f(z+p)=g(z)f(z).$

In the special case where $g$ is identically equal to $1$, we call $f$ a periodic function, 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 $f(z)=z^{2}+1$ quasiperiodic even though it meets the criterion of the definition if we set $g(z)=(z^{2}+2z+2)/(z^{2}+1)$ because the rational function $g$ is “more complicated” than the polynomial $f$. On the other hand, for the gamma function, one would say that $1$ is a quasiperiod because $\Gamma(z+1)=z\Gamma(z)$ and the function $g(z)=z$ is a “much simpler” function than the gamma function.

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

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