[1] Several experiments were made to this end in the United States
(see COMMUNISM) by American followers of Fourier, whose doctrines
were introduced there by Albert Brisbane (1809-1890). Indeed, in the
years between 1840 and 1850, 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.).
FOURIER, JEAN BAPTISTE JOSEPH (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 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 1801, 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 Academie 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 dans 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
_eloges_ of distinguished men of science. His works have been collected
and edited by Gaston Darboux with the title _Oeuvres 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
[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 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 [f](c + [epsilon]), [f](c - [epsilon]) have
distinct but definite limiting values as [epsilon] 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 + 0), [f](c - 0). The discontinuity consists therefore of a
sudden change of value of the function from [f](c - 0) to [f](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 [f](+0), [f](-0).
Suppose we have an infinite series u1(x) + u2(x) + ... + u_n(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 [f](c), the
series [Sigma] [1 to [oo]] u_n(x) is said to represent the function
([f]x) 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 S_n(c),
the condition that S_n(c) converges to the value [f](c) is that,
corresponding to any finite positive number [delta] as small as we
please, a value n1 of n can be found such that if n >= n_1, |[f](c) -
Sn(c)| < [delta].
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 [delta], as small as we please, a finite number n1 can
be found such that the arithmetical value of S(x) - S_n(x), where n =>
n1 is less than [delta] 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) - S_n(x)| may be
< [delta], 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 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[<=]l is capable of
being represented in general by an infinite series of the form
[pi]x 2[pi]x n[pi]x
A1 sin ----- + A2 sin ------ + ... + A_n sin ------ + ...,
l l l
and if it be further assumed that the series is in general uniformly
convergent throughout the interval 0 to l, the form of the
coefficients A can be determined. Multiply each term of the series by
sin n[pi]x/l, and integrate the product between the limits 0 and l,
then in virtue of the property [int][l to 0] sin n[pi]x/l sin
n'[pi]x/l dx=0, or 1/2 l, according as n' is not, or is, equal to n,
we have -1/2 lA_n= [int][0 to l] [f](x) sin n[pi]x/l dx, and thus the
series is of the form
_
2 [oo] n[pi]x / l n[pi]x
-- [Sigma] sin ------ | [f](x) ------ dx. (1)
l 1 l _/ 0 l
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 [integral][0 to
l] [f](x)sin n[pi]x/l dx has a definite meaning for every value of n.
Before we proceed to examine the justification for the assumptions
made, it is desirable to examine the result obtained, and to deduce
other series from it. In order to obtain a series of the form
[pi]x 2[pi]x n[pi]x
B0 + B1 cos ----- + B2 cos ------ + ... + B_n cos ------ + ...
l l l
for the representation of [f](x) in the interval 0 to l, let us apply
the series (1) to represent the function [f](x) sin [pi]x/l; we thus
find
_
2 [oo] n[pi]x / l [pi]x n[pi]x
-- [Sigma] sin ------ | [f](x)sin ----- sin ------ dx,
l 1 l _/ 0 l l
or
_ _ _
1 [oo] n[pi]x / l | (n - 1)[pi]x (n + 1)[pi]x |
-- [Sigma] sin ------ | [f](x) | cos ------------ - cos ------------ | dx.
l 1 l _/ 0 |_ l l _|
On rearrangement of the terms this becomes
_ _
1 [pi]x / l 2 [pi]x n[pi]x / l n[pi]x
-- sin ----- | [f](x) dx + -- [Sigma] sin ----- cos ------ | [f](x) cos ------ dx.
l l _/ 0 l l l _/ 0 l
hence [f](x) is represented for the interval 0 to l by the series of cosines
_ _
1 / l 2 [oo] n[pi]x / l n[pi]x
-- | [f](x) dx + -- [Sigma] cos ------ | [f](x) cos ------ dx ... (2)
l _/ 0 l 1 l _/ 0 l
We have thus seen, that with the assumptions made, the arbitrary
function [f](x) may be represented, for the given interval, either by
a series of sines, as in (1), or by a series of cosines, as in (2).
Some important differences between the two series must, however, be
noticed. In the first place, the series of sines has a vanishing sum
when x=o or x=l; it therefore does not represent the function at the
point x=o, unless [f](0) = 0, or at the point x=l, unless [f](l) = 0,
whereas the series (2) of cosines may represent the function at both
these points. Again, let us consider what is represented by (1) and
(2) for values of x which do not lie between 0 and l. As [f](x) is
given only for values of x between 0 and l, the series at points
beyond these limits have no necessary connexion with [f](x) unless we
suppose that [f](x) is also given for such general values of x in such
a way that the series continue to represent that function. If in (1)
we change x into -x, leaving the coefficients unaltered, the series
changes sign, and if x be changed into x + 2l, the series is
unaltered; we infer that the series (1) represents an odd function of
x and is periodic of period 2l; thus (1) will represent [f](x) in
general for values of x between [+-][oo], only if [f](x) is odd and
has a period 2l. If in (2) we change x into -x, the series is
unaltered, and it is also unaltered by changing x into x + 2l; from
this we see that the series (2) represents [f](x) for values of x
between [+-][oo], only if [f](x) is an even function, and is periodic
of period 2l. In general a function [f](x) arbitrarily given for all
values of x between [+-][oo] is neither periodic nor odd, nor even,
and is therefore not represented by either (1) or (2) except for the
interval 0 to l.
From (1) and (2) we can deduce a series containing both sines and
cosines, which will represent a function [f](x) arbitrarily given in
the interval -l to l, for that interval. We can express by (1) the
function -1/2{[f](x) - [f](-x)} which is an odd function, and thus this
function is represented for the interval -l to +l by
_
2 n[pi]x / l n[pi]x
-- [Sigma] sin ------ | 1/2 {[f](x) - [f](-x)} sin ------ dx;
l l _/ 0 l
we can also express 1/2 {[f](x) + [f](-x)}, which is an even function,
by means of (2), thus for the interval -l to +l this function is
represented by
_ _
1 / l 2 [oo] n[pi]x / l n[pi]x
-- | 1/2 {[f](x) + [f](-x)} dx + -- [Sigma] cos ------ | 1/2 {[f](x) + [f](-x)} cos ------ dx.
l _/ 0 l 1 l _/ 0 l
It must be observed that [f](-x) is absolutely independent of [f](x),
the former being not necessarily deducible from the latter by putting
-x for x in a formula; both [f](x) and [f](-x) are functions given
arbitrarily and independently for the interval 0 to l. On adding the
expressions together we obtain a series of sines and cosines which
represents [f](x) for the interval -l to l. The integrals
_ _
/ l n[pi]x / l n[pi]x
| [f](-x) cos ------ dx, | [f](-x) sin ------ dx
_/ 0 l _/ 0 l
are equivalent to
_ _
/-l n[pi]x /-l n[pi]x
- | [f](x) cos ------ dx, + | [f](x) sin ------ dx,
_/0 l _/ 0 l
thus the series is
_ _ _
1 / l 1 [oo] n[pi]x / l n[pi]x 1 [oo] n[pi]x / l n[pi]x
-- | [f](x)dx + -- [Sigma] cos ------ | f(x) cos ------ dx + -- [Sigma] sin ------ | [f](x) sin ------ dx,
2l_/-l l 1 l _/-l l l 1 l _/-l l
which may be written
_ _
1 / l 1 [oo] / l n[pi](x - x')
-- | [f](x') dx' + -- [Sigma] | [f](x') cos ------------- dx'. (3)
2l_/-l l 1 _/-l l
The series (3), which represents a function [f](x) arbitrarily given
for the interval -l to l, is what is known as Fourier's Series; the
expressions (1) and (2) being regarded as the particular forms which
(3) takes in the two cases, in which [f](-x) = -[f](x), or [f](-x) =
f(x) respectively. The expression (3) does not represent f(x) at
points beyond the interval -l to l, unless [f](x) has a period 2l. For
a value of x within the interval, at which [f](x) is discontinuous,
the sum of the series may cease to represent [f](x), but, as will be
seen hereafter, has the value 1/2 {[f](x + 0) + [f](x - 0)}, the mean
of the limits at the points on the right and the left. The series
represents the function at x=o, unless the function is there
discontinuous, in which case the series is 1/2 {[f](+0) + [f](-0)};
the series does not necessarily represent the function at the points l
and -l, unless [f](l) = [f](-l). Its sum at either of these points is
1/2 {[f](l) + [f](-l)}.
_Examples of Fourier's Series._--(a) Let [f](x) be given from 0 to l,
by [f](x)=c, when 0 <= x < 1/2 l, and by f(x)= -c from 1/2 l to l; it
is required to find a sine series, and also a cosine series, which
shall represent the function in the interval.
We have
_ _ _
/ l n[pi]x /1/2 l n[pi]x / l n[pi]x
| [f](x) sin ------ dx = c | sin ------ dx - c | sin ------ dx
_/ 0 l _/ 0 l _/1/2 l l
cl
= ----- (cos n[pi] - 2 cos 1/2 n[pi] + 1).
n[pi]
This vanishes if n is odd, and if n = 4m, but if n = 4m + 2 it is
equal to 4cl/n[pi]; the series is therefore
4c /l 2[pi]x 1 6[pi]x 1 10[pi]x \
---- ( -- sin ------ + -- sin ------ + -- sin ------- + ... ).
[pi] \2 l 3 l 5 l /
For unrestricted values of x, this series represents the ordinates of
the series of straight lines in fig. 1, except that it vanishes at the
points 0, (1/2)l, l, (3/2)l ...
[Illustration: FIG. 1.]