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Originally appearing in Volume V10, Page 757 of the 1911 Encyclopedia Britannica.
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JEAN BAPTISTE JOSEPH FOURIER (1768-1830), French mathematician, was born at Auxerre on the 21st of March 1768. He was the son of a tailor, and was left an orphan in his eighth year; but, through the kindness of a friend, admission was gained for him into the military school of his native town, which was then under the direction of the Benedictines of Saint-Maur. He soon distinguished himself as a student and made rapid progress, especially in mathematics. Debarred from entering the army on account of his lowness of birth and poverty, he was appointed ' Several experiments were made to this end in the United States (see CoSIMUNISM) by American followers of Fourier, whose doctrines were introduced there by Albert Brisbane (1809-1890). Indeed, in the years between 184o and 185o, during which the movement waxed and waned, no fewer than forty-one phalanges were founded, of which some definite record can be found. The most interesting of all the experiments, not alone from its own history, but also from the fact that it attracted the support of many of the most intellectual and cultured Americans was that of Brook Farm (q.v.). professor of mathematics in the school in which he had been a pupil. In 1787 he became a novice at the abbey of St Benoit-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 Ecole Normale at Paris in 1795 he was sent to teach in it, and was afterwards attached to the Ecole 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 Kleber and Desaix. He returned to France in 18o1, and in the following year he was nominated prefect of Isere, 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'Egypte 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 Rhone, was soon after again deprived. He now settled at Paris, was elected to the Academic des Sciences in 1816, but in consequence of the opposition of Louis XVIII. was not admitted till the following year, when he succeeded the Abbe 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 Ecole 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 Academie des Sciences with a memoir in two parts, Theorie des mouvements de la chaleur clans les corps solides. The first part was republished in 1822 as La Theorie 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 equations indeterminees (1831), which contains much original matter. In addition to the works above mentioned, Fourier wrote many memoirs on scientific subjects, and 'loges of distinguished men of science. His works have been collected and edited by Gaston Darboux with the title Euvres de Fourier (Paris, 1889-189o). 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 areconsequently 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 h; 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 f(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 a 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 f(c) as the value of the function assigned to the point c. For any number of points between a and b the function may he 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 f(c+e), f(c—e) have distinct but definite limiting values as c 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 f(c+o), f(c—o). The discontinuity consists therefore of a sudden change of value of the function from f(c—o) to f(c+o), as x increases through the value c. If there is such a discontinuity at the point x=o, we may denote the limits on the right and on the left respectively by f(+o), f(—o). Suppose we have an infinite series a1 (x) +u2(x)+... +u,,,(x)+.. . in which each tennis 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 gc b the sum exists and agrees with the value of f(c), the series '±'u„ (x) is said to represent the function(fx) 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 Se(c), the condition that S,,(c) converges to the value f(c) is that, corresponding to any finite positive number 5 as small as we please, a value in of n can be found such that if n~nh f(c)—S, (c)I<5. 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)+...+ u2(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 5, as small as we please, a finite number n, can be found such that the arithmetical value of S(x) —S,,(x), where n n1 is less than 5, 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) —Se(x) may be <5, in-creases 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 con-verges 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 uniform, but non-uniformity of convergence of the series does not necessarily imply discontinuity in the sum. Form of Fourier's Series.—If it be assumed that a function f(x) arbitrarily given for values of x such that o x For unrestricted values of x, this series represents the ordinates of the series of straight lines in fig. I, except that it vanishes at the points o, 21, 1, g1 .. . -21 -1 0 ?1 We find similarly that the same function is represented by the series 4c ( ax 1 3ix+1 cos 5zrx- a cos l' cos -1--+5 -7- ) during the interval o to 1; for general values of x the series repre- sents the ordinate of the broken line in fig. 2, except that it vanishes at the points 21, l . . -21 =1 b 7 FIG. 2. (b) Let f(x)=x from o to 21, and f(x)=l-x, from 21 to 1; then 1 flax Ii1 max f f(x) sin l dx= fox sin l dx+ f ~l(l-x) sin nidx z z / 2t la cos 2 +n—212„-2 sin 2 na (cos -cos nal 12 12 na l2 na 212 na + na cos na-`na cos 2 +n,,r2 sin 2 =-2a2 sin 2 ing as n' is not, or is, equal to n, we have 2lA,, = f of (x) sin-1 xdx, and thus the series is of the form Z 2 sin-lx f f(x) sin-,xdx . . . (I) This method of determining the coefficients in the series would not be valid without the assumption that the series is in general uniformly convergent, for in accordance with a known theorem the sum of the integrals of the separate terms of the series is otherwise not necessarily equal to the integral of the sum. This assumption being made, it is further assumed that f (x) is such that f of(x)sinnl xdx or hence the sine series is ~l / nx 1 3xx 1 5rx 1 az `sin -3zsin 1 +z sin-~-- ... J For general values of x, the series represents the ordinates of the row of broken lines in fig. 3. The cosine series, which represents the same function for the interval o to 1, may be found to be 1, 21 r 2rx 1 6rx 1 10arx 4_- `cos l +3zcos l +5zcos l + ... J This series represents for general values of x the ordinate of the set of broken lines in fig. 4. Dirichlet's Integral.—The method indicated by Fourier, but first carried out rigorously by Dirichlet, of proving that, with certain restrictions as to the nature of the function f(x), that function is in general represented by the series (3), consists in finding the sum of n+l terms of that series, and then investigating the limiting value of the sum, when n is increased indefinitely. It thus appears that the series is convergent, and that the value towards which its sum converges is if f(x+o)+f(x-o)}, which is in general equal to f(x). It will be convenient throughout to take -s to a as the given interval; any interval -1 to l may be reduced to this by changing x into lx/a, and thus there is no loss of generality. We find by an elementary process that a+cos (x-x') + cos 2(x-x')+ ... + cos n(x-x') sin2tt2 I (x'-x) =2 sin 1(x' -x) Hence, with the new notation, the sum of the first n+I terms of (3) is x 2n2 I (x'- x) ~f~ f( ,)sin 2 sin %(x'-x) dx'. If we suppose f(x) to be continued beyond the interval —ir to a, in such a way that f(x)=f(x+2a), we may replace the limits in this integral by x+a, x—a respectively; if we then put x'-x=2z, and let f(x') =F(z), the expression becomes f _~ F(z)ssin zzdz, where m=2n+I; this expression may be written in the form 1 f'; sin zd 1 sin mz 7^-J o F(z) sin mz z+ f o*F( z) sin z dz .... (4) We require therefore to find the limiting value, when m is indefinitely increased, off o F(z)ss n zzdz ; the form of the second integral being essentially the same. This integral, or rather the slightly more general onej oF(z)ssin zzdz, when 0< is known as Dirichlet's integral. If we write X(z) =F(z) the integral sIn z becomes f hX(z)si Zmz dz, which is the form in which the integral is frequently considered. The Second Mean-Value Theorem.—The limiting value of Dirichlet's integral may be conveniently investigated by means of a theorem in the integral calculus known as the second mean-value theorem. Let a, b be two fixed finite numbers such that aG(x)+f(a+o), x(x) are continuous and never diminish as x increases; the same reasoning 756 applies to every continuous portion of f(x), for which the functions 0(x), x(x) are formed in the same manner; we now take fl(x) = 1(x)+ f(a+o) +C, f2 (x) = x(x) +C, where C is constant between consecutive discontinuities, but may have different values in the next interval between discontinuities; the C can be so chosen that neither 1(x) nor f2(x) diminishes as x increases through a value for which f(x) is discontinuous. We thus see that f(x) =f1(x) f2(x), where fi(x), f2(x) never diminish as x increases from a to b, and are discontinuous only where f(x) is so. The function f(x) is a particular case of a class of functions defined and discussed by Jordan, under the name " functions with limited variation " (functions a variation bornee); in general such functions have not necessarily only a finite number of maxima and minima. Proof of the Convergence of Fourier's Series.—It will now be assumed that a function f(x) arbitrarily given between the values ~r and +7r, has the following properties: (a) The function is everywhere numerically less than some fixed positive number, and continuous except for a finite number of values of the variable, for which it may be ordinarily discontinuous. (b) The function only changes from increasing to diminishing or vice versa, a finite number of times within the interval; this is usually expressed by saying that the number of maxima and minima is finite. These limitations on the nature of the function are known as Dirichlet's conditions; it follows from them that the function is integrable throughout the interval. On these assumptions, we can investigate the limiting value of Dirichlet's integral; it will be necessary to consider only the case of a function F(z) which does not diminish as z increases from o to 4w, since it has been shown that in the general case the difference of two such functions may be taken. The following lemmas will be required: i. Since J o sin zz dz =J o 11+2 cos 2z+2 cos 4z+... +2 cos 2nz}dz = 2 ; this result holds however large the odd integer m may be. 2. IfoO, f a si dO cannot exceed For by the mean-value theorem If k s' Bde l< Q +., a hence I Lh=ee f h sin a; a in particular if a 5 x, f sl B BdB 2 2 Again da f: siBdO_/ -slam a>0, therefore sl B BdO increases as a diminishes, when B < a < 2r ; but Ern a o f a si BdB= , hence sin 0,10 I 1(z), where x(c), ,'(c) are finite, and o < o <1. It is thus seen that fix) may have a finite number of infinities within the given interval, provided the function is integrable through any one of these points; the function is in that case still representable by Fourier's Series. The Ultimate Values of the Coefficients in Fourier's Series.—If f(x) is everywhere finite within the given int¢'eval -2r to +vr, it can be shown that an, b,,, the coefficients of cos nx, sin nx in the series which represent the function, are such that na,,, nb,,, however. written in the form F(0)fa sinn mzdz+ f o{F(z)-F(0)}ss n mzdz m +P µ {F(z)-F(0)1 ssin zzdz where µ is a fixed, number as small as we please; hence if we use lemma (I), and apply the second mean-value theorem, o (z)s. zzdz- 2F(0) = f o {F(z)-F(0)}sin z sin mzaz f~ sin n sin nE ffi(x) cos nxdx =fi( +0) n- +fi(r-0) n with a similar expression, with f2(x) for f,(x), being between s-and -is; the result then follows at once, and is obtained similarly foothe other coefficient. a +nirf of'(x)cos? dx where a represent the points where f(x) is discontinuous. Hence if f(x) is represented by the series Ea„ sin nix, and f'(x) by the series Eb„ cos T ,we have the relation bn= `tea",-i [f(+0) =f (l-0) + E cos na{ f(a+0) f(a-0)}] hence only when the function is everywhere continuous, and f(+o), f(l-o) are both zero, is the series which represents f'(x) obtained at once by differentiating that which represents f(x). The form of the coefficient as discloses the discontinuities of the function and of its differential coefficients, for on continuing the integration by parts we find a„=n [f(+0) ref(1-0) + cosnla{ 91-0)}] +n z[f'(+0) ref'(l-0) + sin-Ti ~f'($+0) f'(0-0)}] +&c. where p are the points at which f'(x) is discontinuous.
End of Article: JEAN BAPTISTE JOSEPH FOURIER (1768-1830)

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