### Topic Page: Fourier, Jean Baptiste Joseph (1768 - 1830)

**Fourier, Jean Baptiste Joseph**from

*Philip's Encyclopedia*

French mathematician and physicist, scientific adviser (1798-1801) to Napoleon in Egypt. Fourier's application of mathematics to the study of heat led him to discover a technique (Fourier analysis) of expressing complex periodic functions in terms of sums of sine and cosine waves. He also developed the Fourier series of sine and cosine functions, which can be used to represent many periodic phenomena.

**VI.25 Jean-Baptiste Joseph Fourier**

*The Princeton Companion to Mathematics*

Unusually for a mathematician, Fourier pursued a distinguished nonmathematical career. He was a civilian member of General Bonaparte’s expedition to Egypt (1798–1801), important enough for the First Consul to make him, in 1802, the Prefect of the département at Grenoble, a position which he held until Emperor Napoleon’s fall in the mid 1810s. Thereafter, Fourier moved to Paris, where he managed to establish himself to the extent of being appointed a secrétaire perpétuel of the Paris Academy of Sciences in 1822.

The prefectureship involved heavy commitments, and Fourier was also active in Egyptology, most notably discovering a teenager named Jean Champollion in Grenoble, who was later to decipher the Rosetta Stone and who helped to found the discipline. Nevertheless, between 1804 and 1815 he also created most of his scientific work. His motivation was the mathematical study of the diffusion of heat in continuous and solid bodies; his “diffusion equation” for this purpose was not only novel in itself but also marked the first large-scale mathematization of physical phenomena that lay outside mechanics. To solve this differential equation he proposed using infinite trigonometric series. These series were already known but had a low status. Fourier (re)found many properties: not only the formulas for their coefficients and some conditions for their convergence but especially their representability, namely, how a periodic series could represent a general function. For diffusion in a cylinder he found many properties of the Bessel function J_{0}(x), which was then little studied.

Fourier presented his findings to the scientific class of the Institut de France in 1807. lagrange [VI.22] did not like the series, while laplace [VI.23] was disappointed in the physical modeling. But Laplace also gave him a clue about solutions of the diffusion equation for infinite bodies, which led Fourier to find, by 1811, his integral solution (including inversion) for them. His main publication was the book Théorie Analytique de la Chaleur (1822), which greatly influenced younger mathematicians: for example, the first satisfactory proof of the convergence of the series by dirichlet [VI.36] (1829) and their use in fluid dynamics by C. L. M. H. Navier (1825). Less happy was his relationship with poisson [VI.27], who tried to rederive the entire theory following the molecularist physical principles of Laplace and the methods of solution of Lagrange, but only added a few special cases.

Fourier also worked on other topics in mathematics. As a teenager he gave the first proof of descartes’s [VI.11] rule of signs on the numbers of positive and negative roots of a polynomial equation. (He used an inductive proof that has now become standard.) He also found an upper bound on the number of roots within a given interval, which J. C. F. Sturm improved to an exact evaluation in 1829. At that time Fourier was trying to finish a book on equations, which appeared posthumously in 1831 thanks to Navier. The main novelty was the basic theory of linear programming [III.84], as we now call it. Despite his prestige and advocacy, he gained few followers (Navier was one), and the theory lay dormant for over a century. Fourier also took up a few aspects of Laplace’s work on mathematical statistics, examining the status of the normal distribution [III.71 §5].

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