# antiperiodic function

A special case of the quasiperiodicity (http://planetmath.org/Period3) of functions^{} is the antiperiodicity.
An antiperiodic function $f$ satisfies for a certain constant $p$ the equation

$$f(z+p)=-f(z)$$ |

for all values of the variable $z$. The constant $p$ is the antiperiod of $f$. Then, $f$ has also other antiperiods, e.g. $-p$, and generally $(2n+1)p$ with any $n\in \mathbb{Z}$.

The antiperiodic function $f$ is always as well periodic with period $2p$, since

$$f(z+2p)=f((z+p)+p)=-f(z+p)=-(-f(z))=f(z).$$ |

Naturally, then there are all periods $2np$ with $n\in \mathbb{Z}$.

Not all periodic functions^{} are antiperiodic.

For example, the sine and cosine functions are antiperiodic with $p=\pi $, which is their absolutely least antiperiod:

$$\mathrm{sin}(z+\pi )=-\mathrm{sin}z,\mathrm{cos}(z+\pi )=-\mathrm{cos}z$$ |

The tangent (http://planetmath.org/Trigonometry) and cotangent functions are not antiperiodic although they are periodic (with the prime period $\pi $; see complex tangent and cotangent).

The exponential function^{} is antiperiodic with the antiperiod $i\pi $ (see Euler relation):

$${e}^{z+i\pi}={e}^{z}{e}^{i\pi}=-{e}^{z}$$ |

Title | antiperiodic function |
---|---|

Canonical name | AntiperiodicFunction |

Date of creation | 2015-12-16 15:19:14 |

Last modified on | 2015-12-16 15:19:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 30A99 |

Related topic | PeriodicFunctions |

Related topic | QuasiperiodicFunction |

Defines | antiperiodicity |

Defines | antiperiodic |

Defines | antiperiod |