Periodic Function
A periodic function is a function that repeats its values at regular intervals, called the period.
Formula
f(x + T) = f(x) \text{ for all } x, \text{ where } T \text{ is the period}
Definition
A periodic function is one whose graph repeats over and over in a regular pattern; after a distance called the period, it looks exactly the same as before. Formally, $f$ is periodic with period $T$ if $f(x + T) = f(x)$ for all $x$ in the domain, and the fundamental period is the smallest positive such $T$; sin and cos have fundamental period $2\pi$, while tan and cot have fundamental period $\pi$. If $f$ is integrable and periodic with period $T$, its Fourier series is $f(x) = \sum c_n e^{2\pi inx/T}$ with $c_n = (1/T)\int_0^T f(x)e^{-2\pi inx/T}\, dx$, converging pointwise at points of continuity under Dirichlet conditions.
Example
The sine function is periodic with period $2\pi$; the seasons repeat with period $1$ year, and a clock hand with period $12$ hours. For $f(x) = \sin(2x) + \cos(3x)$: $\sin(2x)$ has period $\pi$ and $\cos(3x)$ has period $2\pi/3$, so the combined period is their least common multiple, $2\pi$. The sawtooth wave $f(x) = x/\pi$ on $(-\pi, \pi)$, extended periodically, has Fourier series $2\sum (-1)^{n+1}\sin(nx)/n$, which converges to $f(x)$ except at discontinuities, where it converges to the midpoint (the Gibbs phenomenon).
Key Insight
Periodic functions are nature's repeating patterns: day and night, heartbeats, pendulum swings, and musical notes are all periodic, and the period of a sum of periodic functions is the LCM of the individual periods, though if the ratio of two periods is irrational, the sum is not periodic at all (for example, $\sin(x) + \sin(\sqrt{2}x)$). Every integrable periodic function can be uniquely decomposed into its Fourier components, the mathematical basis for everything from MP3 audio compression to the periodic energy eigenstates of quantum mechanics.