The Fourier series is a representation of a function as an infinite sum of sinusoids.
June 21, 2017—Michael Dobbs
The Fourier Series
History
Jean-Baptiste Joseph Fourier introduced the Fourier series as a way of solving the heat equation in a metal plate. In turn, he concluded that any arbitrary continuous function can be represented by a trigonometric series based on the set of
Create the fourth-order Fourier series approximation:
Plot it:
Create the seventh-order Fourier series approximation:
Plot it:
Application: Numerical Analysis
We can then compare the numerical integral to the integral of the Fourier series (of which all the terms are integrable).
Compute a list of differences between the integrals as the order of the series increases and plot:
We can see that the difference is tending toward 0. Thus, taking the integral of the Fourier series of a function is one way to approximate the area under a curve.
Another Application: Solving Differential Equations
First find the Fourier series of the right-hand side of the equation. We show the first few terms here:
To begin, we compute the second derivative on the left-hand side of the equation for each term:
We then solve for the unknown coefficients, after generalizing the coefficients on the right-hand side. For the even terms:
For the odd terms:
FURTHER EXPLORATIONS
Explore the Taylor Series Expansion of a Function
Explore Other Sets of Basis Functions That Can Approximate a Function