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.