Chapter 2

Periodic Functions

Q: What is a periodic function?

A function $f(x)$ is called a periodic function with period $T$ when there exists a positive constant $T$ such that $f(x+T) = f(x)$ holds for all $x$. The smallest positive such $T$ is called the fundamental period. Typical examples are $\sin x$ (with $T = 2\pi$) and $\tan x$ (with $T = \pi$).

Q: What is the trick to drawing the graph of a periodic function?

Draw the shape over a single period (for example $0 \leq x < T$) and repeat it to the left and right. Check the zeros (intersections with the $x$-axis) and the positions of the maxima and minima, and explicitly indicate jump points when the function is discontinuous. When drawing a sum of periodic functions, superimpose each component.

Q: How are non-periodic functions handled in Fourier analysis?

On a finite interval $[a, b]$, the function is extended as a periodic function of period $T = b - a$ and a Fourier series is used. On the whole real line $(-\infty, \infty)$, one uses the Fourier transform (a continuous spectrum), obtained as the limit $T \to \infty$.

Functions with repeating patterns

Introduction (High School Level)

Introduction

Many phenomena in nature follow a repeating pattern: the heartbeat, the orbit of the Earth, the waveform of alternating current — all of these are "periodic" phenomena.

This chapter studies the "periodic functions" used to describe such phenomena.

Definition of a Periodic Function

Definition: periodic function

A function $f(x)$ is said to be a periodic function if there exists a positive constant $T > 0$ such that, for every $x$,

$$f(x + T) = f(x)$$

holds. In this case $T$ is called a period of $f$.

Fundamental period

The period of a periodic function is not unique: if $T$ is a period, so are $2T, 3T, \ldots$. The smallest positive period is called the fundamental period.

Example: periods of trigonometric functions

The fundamental period of $\sin x$ and $\cos x$ is $2\pi$.
The fundamental period of $\tan x$ is $\pi$.
The fundamental period of $\sin(2x)$ is $\pi$ (doubling the frequency halves the period).

Parameters of a Periodic Function

A general sinusoidal wave can be written in the form:

$$f(x) = A\sin(\omega x + \phi)$$

Amplitude $A$

This represents the "height" of the wave. $|A|$ equals the difference between the maximum and the mean (and the absolute value of the difference between the minimum and the mean).

Angular frequency $\omega$

This represents "how fast the wave oscillates." Its relation to the period $T$ is:

$$T = \dfrac{2\pi}{\omega}$$

Phase $\phi$

This represents the "shift" of the wave. Changing the phase shifts the entire waveform to the left or right.

Example

For $f(x) = 3\sin(2x + \dfrac{\pi}{4})$:

Amplitude: $A = 3$
Angular frequency: $\omega = 2$
Period: $T = \dfrac{2\pi}{2} = \pi$
Phase: $\phi = \dfrac{\pi}{4}$

Frequency

In physics and engineering one often uses the reciprocal of the period, the frequency:

$$f = \dfrac{1}{T} = \dfrac{\omega}{2\pi}$$

The unit of frequency is the Hz (hertz), which expresses "how many oscillations per second."

Example: sound

The reference pitch A (concert A) has a frequency of 440 Hz.
Going up one octave doubles the frequency.

Examples of Periodic Functions

Square wave

This alternates between $+1$ and $-1$ at regular intervals:

$$f(x) = \begin{cases} 1 & (0 \leq x < \pi) \\ -1 & (\pi \leq x < 2\pi) \end{cases}$$

and is repeated with period $2\pi$.

Square wave

Triangle wave

This rises and falls linearly:

$$f(x) = |x| \quad (-\pi \leq x < \pi)$$

and is repeated with period $2\pi$.

Triangle wave

Sawtooth wave

This rises linearly and then drops abruptly:

$$f(x) = x \quad (-\pi \leq x < \pi)$$

and is repeated with period $2\pi$.

Sawtooth wave

These non-sinusoidal periodic functions can also be expressed as sums of sinusoidal waves through Fourier series. This is the power of Fourier analysis.

Summary

A periodic function is a function satisfying $f(x+T) = f(x)$.
The parameters of a sinusoidal wave: amplitude, angular frequency (or period), and phase.
Frequency is the reciprocal of the period.
Square, triangle, and sawtooth waves are also periodic functions.

Frequently Asked Questions

Q1. What is a periodic function?

A: A function $f(x)$ is called a periodic function with period $T$ when there exists a positive constant $T$ such that $f(x+T) = f(x)$ holds for all $x$. The smallest positive such $T$ is called the fundamental period. Typical examples are $\sin x$ (with $T = 2\pi$) and $\tan x$ (with $T = \pi$).

Q2. What is the trick to drawing the graph of a periodic function?

A: Draw the shape over a single period (for example $0 \leq x < T$) and repeat it to the left and right. Check the zeros (intersections with the $x$-axis) and the positions of the maxima and minima, and explicitly indicate jump points when the function is discontinuous. When drawing a sum of periodic functions, superimpose each component.

Q3. How are non-periodic functions handled in Fourier analysis?

A: On a finite interval $[a, b]$, the function is extended as a periodic function of period $T = b - a$ and a Fourier series is used. On the whole real line $(-\infty, \infty)$, one uses the Fourier transform (a continuous spectrum), obtained as the limit $T \to \infty$.