Today in History – December 21, 1807 – Fourier introduces his series at the Paris Institute. Joseph Fourier’s memoir, On the Propagation of Heat in Solid Bodies, was read to the Paris Institute. It introduced the expansion of functions into trigonometric series which are now called Fourier series.

The Fourier series allows periodic functions to be represented as a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. The weights, or coefficients, of the components, arranged in order of increasing frequency, form a sequence (or function) called Fourier series. Fourier analysis provides a frequency domain representation of a time domain function. The mapping between the two functions is one-to-one, so the transform is reversible. A common visualization of this transformation is the audio equalizer, which is a dynamic representation of a time signal converted to the frequency domain. An audio spectrum of both time and frequency is shown below.

Preliminary work by Madhava, Nilakantha Somayaji, Jyesthadeva, Leonhard Euler, Jean le Rond d’Alembert, and Daniel Bernoulli would serve as the foundation for Fourier’s work. He applied his studies of trigonometric series to a solution of the partial differential heat equation to produce the series below:

Fourier’s initial series lacked the precision of a function, and Dirichlet and Riemann would later express the series as a formal integral.

Fourier series applications include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. Using the tools and techniques of spectroscopy, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star’s emitted light. Similarly, engineers can optimize the design of a telecommunications system using information about the spectral components of the data signal that the system will carry.

For more information, see the Engineering Pathway‘s resources on Fourier and the Fourier series For related educational resources, visit the Electrical Engineering Education disciplinary community.