Fourier, Jean Baptiste Joseph

Born Mar. 21, 1768, in Auxerre; died May 16, 1830, in Paris. French mathematician. Member of the Académie des Sciences (1817).

After graduating from the military school in Auxerre, Fourier became a teacher at the school. From 1796 to 1798 he taught at the Ecole Polytechnique.

Fourier’s first works were on algebra. In 1796 he taught his students at the Ecole Polytechnique a theorem, now known as Fourier’s theorem, on the number of real roots of an algebraic equation that lie between given boundaries; the theorem was published in 1820. A complete solution of the problem of the number of real roots of an algebraic equation was obtained by J. C. F. Sturm in 1829. In 1818, Fourier investigated the problem of the conditions of applicability of I. Newton’s method for the numerical solution of equations. Results similar to Fourier’s had been obtained in 1768 by the French mathematician J. R. Mourraille; Fourier, however, did not know of his predecessor’s findings. Fourier’s work on numerical methods of the solution of equations was presented in The Analysis of Determinate Equations, which was published posthumously in 1831.

Fourier’s most important contributions were made in mathematical physics. In 1807 and 1811 he presented his first discoveries in the theory of heat propagation in a solid to the Académie des Sciences. In 1822 he published his famous work The Analytical Theory of Heat, which played an important role in the subsequent history of mathematics. In this work Fourier derived the differential equation of heat conduction and developed ideas that had been outlined by D. Bernoulli. In addition, he worked out the method of separation of variables for solving the equations of heat conduction under various boundary conditions and applied the method to a number of particular cases, such as the cube and cylinder. The method is based on the representation of functions by the trigonometric series now called Fourier series. Although such series had been considered previously, they did not become an effective and important tool of mathematical physics until Fourier used them.

The method of separation of variables was developed further by S. Poisson, M. V. Ostrogradskii, and other 19th-century mathematicians. The Analytical Theory of Heat was the starting point for the creation of the theory of trigonometric series and for the elaboration of some general problems of mathematical analysis. Fourier gave the first examples of the expansion in Fourier series of functions that are defined in different regions by different analytic expressions. He thereby made an important contribution to the resolution of the dispute over the concept of function that had involved the most prominent mathematicians of the 18th century. His attempt to prove the possibility of expanding any function in a Fourier series was unsuccessful but gave rise to an important group of studies devoted to the problem of the representability of functions by trigonometric series. P. Dirichlet, N. I. Lobachevskii, and B. Riemann were among the mathematicians who investigated the problem. The development of set theory and the theory of functions of a real variable was largely based on these studies.