professor of mathematics in the school in which he had been a pupil. In 1787 he became a novice at the abbey of St Benoît-sur-Loire; but he left the abbey in 1789 and returned to his college, where, in addition to his mathematical duties, he was frequently called to lecture on other subjects,—rhetoric, philosophy and history. On the institution of the École Normale at Paris in 1795 he was sent to teach in it, and was afterwards attached to the École Polytechnique, where he occupied the chair of analysis. Fourier was one of the savants who accompanied Bonaparte to Egypt in 1798; and during this expedition he was called to discharge important political duties in addition to his scientific ones. He was for a time virtually governor of half Egypt, and for three years was secretary of the Institut du Caire; he also delivered the funeral orations for Kléber and Desaix. He returned to France in 1801, and in the following year he was nominated prefect of Isère, and was created baron and chevalier of the Legion of Honour. He took an important part in the preparation of the famous Description de l’Égypte and wrote the historical introduction. He held his prefecture for fourteen years; and it was during this period that he carried on his elaborate and fruitful investigations on the conduction of heat. On the return of Napoleon from Elba, in 1815, Fourier published a royalist proclamation, and left Grenoble as Napoleon entered it. He was then deprived of his prefecture, and, although immediately named prefect of the Rhône, was soon after again deprived. He now settled at Paris, was elected to the Académie des Sciences in 1816, but in consequence of the opposition of Louis XVIII. was not admitted till the following year, when he succeeded the Abbé Alexis de Rochon. In 1822 he was made perpetual secretary in conjunction with Cuvier, in succession to Delambre. In 1826 Fourier became a member of the French Academy, and in 1827 succeeded Laplace as president of the council of the École Polytechnique. In 1828 he became a member of the government commission established for the encouragement of literature. He died at Paris on the 16th of May 1830.
As a politician Fourier achieved uncommon success, but his fame chiefly rests on his strikingly original contributions to science and mathematics. The theory of heat engaged his attention quite early, and in 1812 he obtained a prize offered by the Académie des Sciences with a memoir in two parts, Théorie des mouvements de la chaleur dans les corps solides. The first part was republished in 1822 as La Théorie analytique de la chaleur, which by its new methods and great results made an epoch in the history of mathematical and physical science (see below: Fourier’s Series). An English translation has been published by A. Freeman (Cambridge, 1872), and a German by Weinstein (Berlin, 1884). His mathematical researches were also concerned with the theory of equations, but the question as to his priority on several points has been keenly discussed. After his death Navier completed and published Fourier’s unfinished work, Analyse des équations indéterminées (1831), which contains much original matter. In addition to the works above mentioned, Fourier wrote many memoirs on scientific subjects, and éloges of distinguished men of science. His works have been collected and edited by Gaston Darboux with the title Œuvres de Fourier (Paris, 1889–1890).
For a list of Fourier’s publications see the Catalogue of Scientific Papers of the Royal Society of London. Reference may also be made to Arago, “Joseph Fourier,” in the Smithsonian Report (1871).
FOURIER’S SERIES, in mathematics, those series which proceed according to sines and cosines of multiples of a variable, the various multiples being in the ratio of the natural numbers; they are used for the representation of a function of the variable for values of the variable which lie between prescribed finite limits. Although the importance of such series, especially in the theory of vibrations, had been recognized by D. Bernoulli, Lagrange and other mathematicians, and had led to some discussion of their properties, J. B. J. Fourier (see above) was the first clearly to recognize the arbitrary character of the functions which the series can represent, and to make any serious attempt to prove the validity of such representation; the series are consequently usually associated with the name of Fourier. More general cases of trigonometrical series, in which the multiples are given as the roots of certain transcendental equations, were also considered by Fourier.
Before proceeding to the consideration of the special class of series to be discussed, it is necessary to define with some precision what is to be understood by the representation of an arbitrary function by an infinite series. Suppose a function of a variable x to be arbitrarily given for values of x between two fixed values a and b; this means that, corresponding to every value of x such that a ≦ x ≦ b, a definite arithmetical value of the function is assigned by means of some prescribed set of rules. A function so defined may be denoted by ƒ(x); the rules by which the values of the function are determined may be embodied in a single explicit analytical formula, or in several such formulae applicable to different portions of the interval, but it would be an undue restriction of the nature of an arbitrarily given function to assume à priori that it is necessarily given in this manner, the possibility of the representation of such a function by means of a single analytical expression being the very point which we have to discuss. The variable x may be represented by a point at the extremity of an interval measured along a straight line from a fixed origin; thus we may speak of the point c as synonymous with the value x = c of the variable, and of ƒ(c) as the value of the function assigned to the point c. For any number of points between a and b the function may be discontinuous, i.e. it may at such points undergo abrupt changes of value; it will here be assumed that the number of such points is finite. The only discontinuities here considered will be those known as ordinary discontinuities. Such a discontinuity exists at the point c if ƒ(c + ε), ƒ(c − ε) have distinct but definite limiting values as ε is indefinitely diminished; these limiting values are known as the limits on the right and on the left respectively of the function at c, and may be denoted by ƒ(c + 0), ƒ(c − 0). The discontinuity consists therefore of a sudden change of value of the function from ƒ(c − 0) to ƒ(c + 0), as x increases through the value c. If there is such a discontinuity at the point x = 0, we may denote the limits on the right and on the left respectively by ƒ(+0), ƒ(−0).
Suppose we have an infinite series u1(x) + u2(x) + . . . + un(x) + . . . in which each term is a function of x, of known analytical form; let any value x = c (a ≦ c ≦ b) be substituted in the terms of the series, and suppose the sum of n terms of the arithmetical series so obtained approaches a definite limit as n is indefinitely increased; this limit is known as the sum of the series. If for every value of c such that a ≦ c ≦ b the sum exists and agrees with the value of ƒ(c), the series is said to represent the function (ƒx) between the values a, b of the variable. If this is the case for all points within the given interval with the exception of a finite number, at any one of which either the series has no sum, or has a sum which does not agree with the value of the function, the series is said to represent “in general” the function for the given interval. If the sum of n terms of the series be denoted by Sn(c), the condition that Sn(c) converges to the value ƒ(c) is that, corresponding to any finite positive number δ as small as we please, a value n1 of n can be found such that if n ≧ n1, |ƒ(c) − Sn(c)| < δ.
Functions have also been considered which for an infinite number of points within the given interval have no definite value, and series have also been discussed which at an infinite number of points in the interval cease either to have a sum, or to have one which agrees with the value of the function; the narrower conception above will however be retained in the treatment of the subject in this article, reference to the wider class of cases being made only in connexion with the history of the theory of Fourier’s Series.
Uniform Convergence of Series.—If the series u1(x) + u2(x) + . . . + un(x) + . . . converge for every value of x in a given interval a to b, and its sum be denoted by S(x), then if, corresponding to a finite positive number δ, as small as we please, a finite number n1 can be found such that the arithmetical value of S(x) − Sn(x), where n ⪚ n1 is less than δ for every value of x in the given interval, the series is said to converge uniformly in that interval. It may however happen that as x approaches a particular value the number of terms of the series which must be taken so that |S(x) − Sn(x)| may be < δ, increases indefinitely; the convergence of the series is then infinitely slow in the neighbourhood of such a point, and the series is not uniformly convergent throughout the given interval, although it converges at each point of the interval. If the number of such points in the neighbourhood of which the series ceases to converge uniformly be finite, they may be excluded by taking intervals of finite magnitude as small as we please containing such points, and considering the convergence of the series in the given interval with such sub-intervals excluded; the convergence of the series is now uniform throughout the remainder of the interval. The series is said to be in general uniformly convergent within the given interval a to b if it can be made uniformly convergent by the exclusion of a finite number of portions of the interval, each such portion being arbitrarily small. It is known that the sum of an infinite series of continuous terms can be discontinuous only at points in the neighbourhood of which the convergence of the series is not
