Periodic function

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Example of a periodic function, with period  T {\displaystyle T} . If you choose any point on the function and then move to the left or right by  T {\displaystyle T} , you will find the same value as at the original point.

In mathematics a periodic function is a function that repeats itself after a while, and indefinitely. The mathematical definition of this is that  f ( t ) {\displaystyle f(t)} is periodic with period T {\displaystyle T} if

f ( t + T ) = f ( t ) ∀ t ∈ R . {\displaystyle f(t+T)=f(t)\ \ \forall \ t\in \mathbb {R} \ .}

Common examples of periodic functions are  sin ⁡ ( ω t ) {\displaystyle \sin(\omega t)} and  cos ⁡ ( ω t ) {\displaystyle \cos(\omega t)} , which both have period  2 π / ω {\displaystyle 2\pi /\omega } .

A sawtooth wave is a periodic function that can be described by

f ( x ) = { | x − 1 | if − 1 < x < 1 , f ( x + 2 ) if x ≤ − 1 , f ( x − 2 ) if x ≥ 1. {\displaystyle f(x)={\begin{cases}|x-1|&{\text{if }}-1<x<1,\\\f(x+2)&{\text{if }}x\leq -1,\\\f(x-2)&{\text{if }}x\geq 1.\end{cases}}}