A function $f$ 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 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.