Definition:Periodic Function

From ProofWiki
Jump to: navigation, search

Definition

Periodic Real Function

Let $f: \R \to \R$ be a real function.


Then $f$ is periodic if and only if:

$\exists L \in \R_{\ne 0}: \forall x \in \R: f \left({x}\right) = f \left({x + L}\right)$


Periodic Complex Function

Let $f: \C \to \C$ be a complex function.


Then $f$ is periodic if and only if:

$\exists L \in \C_{\ne 0}: \forall x \in \C: f \left({x}\right) = f \left({x + L}\right)$


Periodic Element

Let $L \in X_{\ne 0}$.

Then $L$ is a periodic element of $f$ if and only if:

$\forall x \in X: \map f x = \map f {x + L}$


Period

The period of $f$ is the smallest value $\cmod L \in \R_{\ne 0}$ such that:

$\forall x \in X: \map f x = \map f {x + L}$

where $\cmod L$ is the modulus of $L$.


Frequency

The frequency $\nu$ of $f$ is the reciprocal of the period $L$ of $f$:

$\nu = \dfrac 1 L$


Also see

  • Results about periodic functions can be found here.