 For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 "Mémoire sur la propagation de la chaleur dans les corps solides" ("Treatise on the propagation of heat in solid bodies"), and publishing his "Théorie analytique de la chaleur" ("Analytical theory of heat") in 1822.