17calculus 17calculus
First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Integrating Factors (Linear)
Exact Equations
Integrating Factors (Exact)
Constant Coefficients
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > fourier series

Topics You Need To Understand For This Page

precalculus - even and odd functions

infinite series

basics of differential equations

Differential Equations Alpha List


Related Topics and Links

external links you may find helpful

fourier series youtube playlist

Fourier Series

This page covers two areas related to Fourier Series. First, we present an introduction to Fourier Series, then we discuss how to solve differential equations using Fourier Series. If you are just learning about Fourier Series, you can go through the introduction and practice problems and skip the section related to solving differential equations.

What is a Fourier Series?

The main idea of Fourier Series is that we want to build an infinite series, using the basic trig functions sine and cosine, that is equivalent to a more complicated function. The series can then be manipulated more easily than the original function.

Here is a great video to get you started. He explains why we need to build these functions, goes through an example and then explains the big picture.

Dr Chris Tisdell - building functions

How to Calculate Fourier Series

As you saw in that video, there are some basic equations required to calculate the Fourier Series.

Fourier Series Equations

original function

build a Fourier Series for a function \(f(t)\) with period \(2L\)

requirements of \(f(t)\)

\(f(t)\) and it's derivative \(f'(t)\) must be piecewise continuous on the interval \([-L,L]\)

Fourier Series

\(\displaystyle{ f(t) = a_0 + \sum_{n=1}^{\infty}{ \left[ a_n \cos \frac{n \pi t}{L} + b_n \sin \frac{n \pi t}{L} \right] } }\)


\(\displaystyle{ a_0 = \frac{1}{2L} \int_{-L}^{L}{f(t)~dt} }\)

\(\displaystyle{ a_n = \frac{1}{L} \int_{-L}^{L}{f(t)\cos \frac{n\pi t}{L} ~dt} }\)

\(\displaystyle{ b_n = \frac{1}{L} \int_{-L}^{L}{f(t)\sin \frac{n\pi t}{L} ~dt} }\)

Knowing if the original \(f(t)\) is either even or odd can help us a lot when finding the Fourier Series. Of course, we do not require that \(f(t)\) be even or odd, but you remember from precalculus that cosine is an even function and sine is odd. So, for even functions \(b_n=0\) and for odd functions \(a_n=0\).

[ more discussion and practice problems on the way ]

Solving Differential Equations

These videos show how to use Fourier Series to solve differential equations.

MIT OCW -Lec 15 | MIT 18.03 Differential Equations, Spring 2006 - Introduction to Fourier Series; Basic Formulas for Period 2(pi)

MIT OCW - Lec 16 | MIT 18.03 Differential Equations, Spring 2006 - Continuation: More General Periods; Even and Odd Functions; Periodic Extension

MIT OCW - Lec 17 | MIT 18.03 Differential Equations, Spring 2006 - Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds

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Practice Problems

Instructions - - Unless otherwise instructed, find the Fourier Series for these functions.

Level A - Basic

Practice A01

\(\displaystyle{ f(t) = \left\{ \begin{array}{rr} -1 & -\pi < t < 0 \\ 0 & t = 0, \pm \pi \\ 1 & 0 < t < \pi \end{array} \right. }\)
\( f(t) = f(t+2\pi)\) for all \(t\)


Practice A02

\(\displaystyle{f(x) = \left\{ \begin{array}{rr} 0 & -1 \leq x \leq 0 \\ 1 & 0 < x < 1 \end{array} \right. }\)
with period \(2\)


Practice A03

\(\displaystyle{ f(x) = \left\{ \begin{array}{rr} -3 & -1 < x < 0 \\ 3 & 0 < x < 1 \end{array} \right. }\)
\( f(x) = f(x+2) \)


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