A function is said to have a quasiperiod if there exists a function such that

In the special case where is identically equal to , we call a periodic function, and we say that is a period of or that has periodicity .

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 . In order for the term “quasiperiodic” not to be trivial, it is customary to reserve its use for the case where the function is, in some vague, intuitive sense, simpler than the function . For instance, no one would call the function quasiperiodic even though it meets the criterion of the definition if we set because the rational function is “more complicated” than the polynomial . On the other hand, for the gamma function, one would say that is a quasiperiod because and the function 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.