# 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 $\mathrm{\Gamma}(z+1)=z\mathrm{\Gamma}(z)$ 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 |