Fourier series rebuild periodic functions from sines and cosines
A Fourier series expands a periodic function as a trigonometric sum. Named for Joseph Fourier, who lived from 1768 to 1830 and attacked the heat equation, the idea had earlier roots in Euler and d'Alembert; Bessel published related work in 1819 before Fourier's 1822 book.
Convergence questions ask how partial sums behave as more terms join, illustrated classically by adding ever more components of a square wave. The Fourier transform generalizes the project to non-periodic functions; on a circle, periodic cases and transform views meet. Modern rigor from Dirichlet and Riemann repaired Fourier's informal early-nineteenth-century arguments.
Heat flow motivated the original attack: before Fourier, general solutions to the heat equation were missing even when special cases were known. Once trigonometric expansions worked there, the same pattern spread to many linear PDE and vibration problems across physics. Fourier's concrete target was heat in a metal plate, and the only known particular solutions were for sources shaped like a sine or cosine wave, now sometimes called eigensolutions.
Fourier found each coefficient by multiplying the target function by one cosine and integrating from minus one to one, because every cross term between different odd cosines cancels over that span; those few lines sit close to the notation used today. Euler, d'Alembert, Daniel Bernoulli and Gauss had used similar trigonometric sums, but Fourier went further and held that such series could represent any arbitrary function. Not everyone was convinced. When he entered a competition essay in 1811, a committee that included Lagrange, Laplace, Malus and Legendre judged that his derivation had difficulties and fell short on generality and rigour. Beyond heat, the method suits linear differential equations with constant coefficients, whose basic solutions are sinusoids, and it found many uses in electrical engineering.
Source: Fourier series
