Fourier Series and Physics

Abstract

Fourier series were first developed to solve partial differential equations that arise in physical problems, such as heat flow and vibration. We will look at the physics problem of heat flow to see how Fourier series arise and why they are useful. Then we will proceed with the solution, which leads to a lot of very interesting mathematics. We will also see that the problem of a vibrating string leads to a different PDE that requires similar techniques to solve.

While these problems sound very applied, the infinite series that arise as solutions forced mathematicians to delve deeply into the foundations of analysis. When d’Alembert proposed his solution for the motion of a vibrating string in 1754, there were no clear, precise definitions of limit, function, or even of the real numbers—all things taken for granted in most calculus courses today. D’Alembert’s solution has a closed form, and thus did not really challenge deep principles. However, the solution to the heat problem that Fourier proposed in 1807 required notions of convergence that mathematicians of that time did not have. Fourier won a major prize in 1812 for this work, but the judges, Laplace, Lagrange, and Legendre, criticized Fourier for lack of rigour. Work in the nineteenth century by many now famous mathematicians eventually resolved these questions by developing the modern definitions of limit, continuity, and uniform convergence. These tools were developed not because of some fetish for finding complicated things, but because they were essential to understanding Fourier series.