Periodic Function¶

Definition¶

Let $D\subseteq\mathbb{R}$ and let $f\colon D\to\mathbb{R}$ be a function. A number $T>0$ is a period of $f$ if, for every $x\in D$$D$, also $x+T\in D$ and $x-T\in D$$D$, and

The function $f$ is periodic if it has a period. The smallest period, if one exists, is the fundamental period. The same definition is used for functions of a complex variable, where the period may be a nonzero complex number.

Interpretation¶

A periodic function repeats itself: its graph on any interval of length $T$ determines the whole graph, by shifting copies $T$ to the left and right. Periodic functions model cycles such as rotations, oscillations and seasonal patterns.

The graph is $\sin(3x)$ from the Examples, whose fundamental period is $T=2\pi/3$$2\pi /3$. The shaded window $[0,T]$ holds one period; every other part of the graph is a copy of it shifted by a multiple of $T$. The two green points sit at some $x$ and at $x+T$ and have the same height, which is the defining equation $f(x+T)=f(x)$$f(x)$.

Key Results¶

• If $T$ is a period of $f$, then so is $kT$ for every positive integer $k$. So "the period" usually means the fundamental period.
• If $T$ is a period of $f$ and $a>0$$0$, then $g(x)=f(ax)$ has period $T/a$, since $g(x+T/a)=f(ax+T)=f(ax)$$f(ax)$.
• Sums and products of functions with a common period $T$ also have period $T$, though their fundamental period can be smaller: $\sin^2x=\tfrac{1}{2}(1-\cos2x)$ has fundamental period $\pi$, not $2\pi$.
• A continuous periodic function on $\mathbb{R}$ is bounded and attains its maximum and minimum: by the extreme value theorem it does so on $[0,T]$, and every value is repeated there.
• A periodic function need not have a fundamental period: a constant function has every $T>0$ as a period, and the Dirichlet function has every positive rational number as a period, but no smallest one.

Examples¶

• $\sin$ and $\cos$ have fundamental period $2\pi$: $\sin(x+2\pi)=\sin x$ and $\cos(x+2\pi)=\cos x$ for all $x\in\mathbb{R}$$\mathbb{R}$.
• $\tan$ has fundamental period $\pi$: shifting by $\pi$ changes the sign of both $\sin x$ and $\cos x$, so $\tan(x+\pi)=\tan x$ wherever $\cos x\neq0$$0$.
• $\sin(3x)$ has fundamental period $2\pi/3$, by the scaling rule above.
• The complex exponential is $2\pi\mathrm{i}$-periodic: by Euler's formula, $\mathrm{e}^{2\pi\mathrm{i}}=\cos2\pi+\mathrm{i}\sin2\pi=1$$1$, so $\mathrm{e}^{z+2\pi\mathrm{i}}=\mathrm{e}^{z}\mathrm{e}^{2\pi\mathrm{i}}=\mathrm{e}^{z}$ for every $z\in\mathbb{C}$$\mathbb{C}$.
• The fractional part $x-\lfloor x\rfloor$ (floor function) has fundamental period $1$.

Prerequisites¶

• Function
• Real Number

Related Concepts¶

• Trigonometric Functions
• Complex Exponential Function
• Euler's Formula
• Dirichlet Function
• Floor and Ceiling Functions

Sources¶

• Math I: Calculus · Lecture 4 (enrichment: supplements the trigonometric functions; not part of the course)

Aliases: period, periodicity Keywords: fundamental period, least period, periodic Symbols: complex numbers, element of, pi, real numbers, subset

Last updated: 17 October 2026 21:53