This is a beta course, so its structure, chapters, and examples may continue to change.

Periodicity of Functions

Definition

Definition of a Periodic Function

If there exists a nonzero constant $T$ such that for every $x$ in the domain, $f(x + T) = f(x)$ , then $f(x)$ is called a periodic function, and $T$ is called a period of $f$ .

How Many Periods Does a Periodic Function Have?

If $T$ is a period, then for any positive integer $n$ , $nT$ is also a period:

f(x + 2T) = f((x + T) + T) = f(x + T) = f(x)

and by induction, $f(x + nT) = f(x)$ for every positive integer $n$ . So a periodic function has infinitely many periods.

Can a Period Be Negative?

Yes. If $T$ is a period, then $-T$ is also a period: letting $y = x - T$ , we have $f(y + T) = f(y)$ , i.e. $f(x) = f(x - T)$ .

For example, $y = \sin x$ repeats every $2\pi$ :

That is, $\sin(x + 2\pi) = \sin x$ for all $x$ .

Fundamental Period

Since any integer multiple $nT$ of a period is also a period, a periodic function has infinitely many positive and negative periods. For convenience we usually care about the fundamental (least positive) period.

Fundamental Period

For a periodic function $f(x)$ , if there exists a positive number $T_0$ such that:

$T_0$ is a period of $f(x)$
no positive number smaller than $T_0$ is a period of $f(x)$

then $T_0$ is called the fundamental period of $f(x)$ .

Meaning:

The fundamental period is the most basic information describing the repetition of the function
Knowing the fundamental period lets you derive all periods ( $nT_0$ , $n$ an integer)

Is the Fundamental Period Unique?

Yes—if a periodic function has a fundamental period, it is unique.

Proof sketch (by contradiction): suppose $T_1$ and $T_2$ are two different fundamental periods with $T_1 > T_2$ . Since both are periods, $T_1 - T_2$ is also a period (because $f(x + (T_1 - T_2)) = f((x + T_1) - T_2) = f(x)$ ). But $0 < T_1 - T_2 < T_1$ , contradicting that $T_1$ is the smallest positive period. Hence the fundamental period is unique.

Is a Constant Function Periodic?

Yes. For $f(x) = c$ , every positive $T$ satisfies $f(x + T) = c = f(x)$ .

However, a constant function has no fundamental period: for any proposed $T_0$ , there is a smaller positive number that is also a period. So not every periodic function has a fundamental period.

Common Periodic Functions

The fundamental periods of the common ones:

$\sin x$ , $\cos x$ : fundamental period $2\pi$
$\tan x$ , $\cot x$ : fundamental period $\pi$
$\sin^2 x$ , $\cos^2 x$ : fundamental period $\pi$

Important Properties

Historical note: these properties were discovered and proved over a long period through the study of trigonometric functions and wave phenomena. Do not worry if they do not feel obvious at first—they are not.

Arithmetic of Periodic Functions

Property: If $f(x)$ and $g(x)$ are periodic with periods $T_1$ and $T_2$ , and they have a common period $T$ (for example, when $T_1/T_2$ is rational), then:

The sum $f(x) + g(x)$ is periodic
The difference $f(x) - g(x)$ is periodic
The product $f(x) \cdot g(x)$ is periodic
The quotient $\frac{f(x)}{g(x)}$ (where defined and the denominator is nonzero) is periodic

Examples:

$\sin x + \cos x$ : both have period $2\pi$ , so the sum has period $2\pi$
$\sin x + \sin(2x)$ : $\sin x$ has period $2\pi$ , $\sin(2x)$ has period $\pi$ ; they share the period $2\pi$ , so the sum has period $2\pi$

Caution: if $T_1$ and $T_2$ are incommensurable, the result of arithmetic may not be periodic at all—for example, $\sin x+\sin(\sqrt{2}x)$ is not periodic.

Composition of Periodic Functions

Property: If $f(x)$ is periodic with period $T$ , the periodicity of $f(g(x))$ depends on the inner function $g(x)$ .

Case 1: if $g(x) = kx + b$ ( $k \neq 0$ ) is linear, then $f(kx + b)$ is periodic with period $\frac{T}{|k|}$ .
Case 2: if $g(x)$ itself is periodic with period $T_g$ , then $f(g(x))$ has $T_g$ as a period; its fundamental period needs further analysis.

Examples:

$f(x) = \sin(3x)$ : since $\sin x$ has period $2\pi$ , $\sin(3x)$ has period $\frac{2\pi}{3}$
$f(x) = \cos(x + \frac{\pi}{4})$ : translation does not change the period, so it is $2\pi$

Integration of Periodic Functions

Note: this part uses definite integrals. If you have not yet studied integration, you can skip it—it will be covered again in the integration-related chapters.

Property: If $f(x)$ is periodic with period $T$ and integrable on $[a, a + T]$ , then its integral over any interval of length one period is the same:

$\mathbb{R}$ (double-struck R): the standard symbol for the set of real numbers.

\int_a^{a+T} f(x) dx = \int_0^T f(x) dx

Meaning: the integral of a periodic function over any full period is identical—useful when computing averages or power of periodic signals.

Example: for $f(x) = \sin x$ with period $2\pi$ ,

\int_a^{a+2\pi} \sin x dx = \int_0^{2\pi} \sin x dx = 0

for every $a$ .

Exercises

Exercise 1

Find the period of the function $f(x) = \sin(3x) + \cos(2x)$ .

Reference Answer(3 个标签)

periodic functionperiod of sum functionsleast common multiple

Solution Approach: We need to find the periods of $\sin(3x)$ and $\cos(2x)$ respectively, then find their least common multiple.

Detailed Steps:

Period of $\sin(3x)$ : $T_1 = \frac{2\pi}{3}$
Period of $\cos(2x)$ : $T_2 = \frac{2\pi}{2} = \pi$
Find the least common multiple: $\frac{2\pi}{3} = \frac{2\pi}{3}$ , $\pi = \frac{3\pi}{3}$ Least common multiple is $2\pi$

Answer: The period of this function is $2\pi$ .

Exercise 2

Determine whether the function $f(x) = \sin^2 x + \cos^2 x$ is a periodic function. If so, find its period.

Reference Answer(3 个标签)

periodic functiontrigonometric identitiesconstant function

Solution Approach: Simplify the function expression using trigonometric identities.

Detailed Steps:

Using the identity: $\sin^2 x + \cos^2 x = 1$
Therefore $f(x) = 1$ , this is a constant function
A constant function is periodic, any non-zero real number is a period
The fundamental period does not exist (because any arbitrarily small positive number is a period)

Answer: This function is periodic but has no fundamental period.

Exercise 3

Find the fundamental period of $f(x) = \tan\left(2x + \frac{\pi}{3}\right)$ .

Reference Answer(3 个标签)

periodic functiontangentcompound function

Solution Approach: Use the composition rule for linear inner functions.

Detailed Steps:

The period of $\tan x$ is $\pi$
The inner function is $g(x) = 2x + \frac{\pi}{3}$ , linear with $k = 2$
By the composition rule, the period is $\frac{T}{|k|} = \frac{\pi}{2}$

Answer: The fundamental period is $\frac{\pi}{2}$ .

Summary

Symbols Used in This Article

Symbol	Type	Reading/Explanation	Meaning in This Article
$f(x)$	Math symbol	f of x	A function of variable $x$
$T$	Math symbol	T	A period of the function
$T_0$	Math symbol	T zero	The fundamental (least positive) period
$n$	Math symbol	n	A positive integer
$\mathbb{R}$	Math symbol	Real numbers	The set of all real numbers
$\pi$	Greek letter	Pi	The constant $\pi \approx 3.14159$
$\int_a^b$	Math symbol	integral from a to b	The definite integral from $a$ to $b$

English–Chinese Glossary

English term	Chinese term	Phonetic	Explanation
periodic function	周期函数	/ˌpɪərɪˈɒdɪk ˈfʌŋkʃən/	A function whose values repeat at fixed intervals
period	周期	/ˈpɪərɪəd/	A constant $T$ with $f(x+T)=f(x)$
fundamental period	最小正周期	/ˌfʌndəˈmentl ˈpɪərɪəd/	The smallest positive period
constant function	常数函数	/ˈkɒnstənt ˈfʌŋkʃən/	A function that always returns the same value
periodicity	周期性	/ˌpɪərɪəˈdɪsəti/	The repeating property of a function
least common multiple	最小公倍数	/liːst ˈkɒmən ˈmʌltɪpl/	The smallest common multiple of several numbers