Periodic Function
TrigonometryA 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 certain distance (called the period), it looks exactly the same as before.
Example
The sine function is periodic with period $2\pi$. The seasons are periodic with period $1$ year. A clock hand is periodic with period $12$ hours.
Key Insight
Periodic functions are nature's repeating patterns. Day and night, heartbeats, pendulum swings, and musical notes are all periodic.
Definition
A function $f$ is periodic with period $T$ if $f(x + T) = f(x)$ for all $x$ in the domain. The fundamental period is the smallest positive $T$ for which this holds. Sin and cos have fundamental period $2\pi$. Tan and cot have fundamental period $\pi$.
Example
$f(x) = \sin(2x) + \cos(3x)$: $\sin(2x)$ has period $\pi$, $\cos(3x)$ has period $2\pi/3$. The combined function's period is the least common multiple of $\pi$ and $2\pi/3 = 2\pi$. So $f(x + 2\pi) = f(x)$.
Key Insight
The period of a sum of periodic functions is the LCM of the individual periods. If the ratio of two periods is irrational, the sum is not periodic at all. For example, $\sin(x) + \sin(\sqrt{2}x)$ is not periodic.
Definition
A periodic function $f: \mathbb{R} \to \mathbb{R}$ with period $T$ is a function invariant under translation by $T$: $f(x + T) = f(x)$. If $f$ is integrable and periodic with period $T$, its Fourier series is $f(x) = \sum c_n e^{2\pi inx/T}$ where $c_n = (1/T)\int_0^T f(x)e^{-2\pi inx/T}\, dx$. Convergence depends on smoothness (Dirichlet conditions guarantee pointwise convergence at points of continuity).
Example
The sawtooth wave $f(x) = x/\pi$ for $x \in (-\pi, \pi)$, extended periodically, has Fourier series $2(\sin(x) - \sin(2x)/2 + \sin(3x)/3 - \ldots) = 2\sum (-1)^{n+1}\sin(nx)/n$. This series converges to $f(x)$ except at discontinuities, where it converges to the midpoint (Gibbs phenomenon occurs near jumps).
Key Insight
Every integrable periodic function can be uniquely decomposed into its Fourier components. This is the mathematical basis for everything from MP3 audio compression (which discards inaudible Fourier components) to quantum mechanics (where energy eigenstates are periodic functions in angle variables).