1911 Encyclopædia Britannica/Fourier's Series

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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 axb, 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 (acb) 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 acb 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 nn1, |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 nn1 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 intervala 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 ƒ(x) arbitrarily given for values of x such that 0 ≦ xl is capable of being represented in general by an infinite series of the form

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 , and integrate the product between the limits 0 and l, then in virtue of the property , or , according as n′ is not, or is, equal to n, we have , and thus the series is of the form

(1)

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 ƒ(x) is such that 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

for the representation of ƒ(x) in the interval 0 to l, let us apply the series (1) to represent the function ; we thus find

or

On rearrangement of the terms this becomes

hence ƒ(x) is represented for the interval 0 to l by the series of cosines

(2)

We have thus seen, that with the assumptions made, the arbitrary function ƒ(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 = 0 or x = l; it therefore does not represent the function at the point x = 0, unless ƒ(0) = 0, or at the point x = l, unless ƒ(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 ƒ(x) is given only for values of x between 0 and l, the series at points beyond these limits have no necessary connexion with ƒ(x) unless we suppose that ƒ(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 ƒ(x) in general for values of x between ±∞, only if ƒ(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 ƒ(x) for values of x between ±∞, only if ƒ(x) is an even function, and is periodic of period 2l. In general a function ƒ(x) arbitrarily given for all values of x between ±∞ 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 ƒ(x) arbitrarily given in the interval −l to l, for that interval. We can express by (1) the function which is an odd function, and thus this function is represented for the interval −l to +l by

we can also express , which is an even function, by means of (2), thus for the interval −l to +l this function is represented by

It must be observed that ƒ(−x) is absolutely independent of ƒ(x), the former being not necessarily deducible from the latter by putting −x for x in a formula; both ƒ(x) and ƒ(−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 ƒ(x) for the interval −l to l. The integrals

are equivalent to

thus the series is

which may be written

(3)

The series (3), which represents a function ƒ(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 ƒ(−x) = −ƒ(x), or ƒ(−x) = ƒ(x) respectively. The expression (3) does not represent ƒ(x) at points beyond the interval −l to l, unless ƒ(x) has a period 2l. For a value of x within the interval, at which ƒ(x) is discontinuous, the sum of the series may cease to represent ƒ(x), but, as will be seen hereafter, has the value , the mean of the limits at the points on the right and the left. The series represents the function at x = 0, unless the function is there discontinuous, in which case the series is ; the series does not necessarily represent the function at the points l and −l, unless ƒ(l) = ƒ(−l). Its sum at either of these points is .

Examples of Fourier’s Series.—(a) Let ƒ(x) be given from 0 to l, by ƒ(x) = c, when 0 ≦ x < 1/2l, and by f(x)= −c from 1/2l 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

This vanishes if n is odd, and if n = 4m, but if n = 4m + 2 it is equal to ; the series is therefore

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, 1/2l, l, 3/2l ...

EB1911 - Fourier's series - Fig. 1.png
Fig. 1.

We find similarly that the same function is represented by the series

during the interval 0 to l; for general values of x the series represents the ordinate of the broken line in fig. 2, except that it vanishes at the points 1/2l, 3/2l ...

EB1911 - Fourier's series - Fig. 2.png
Fig. 2.

(b) Let ƒ(x) = x from 0 to 1/2l, and f(x) = lx, from 1/2l to l; then

hence the sine series is

4l ( sin nx 1 sin 3πx + 1 sin 5πx − . . . )
π2 l 32 l 52l

For general values of x, the series represents the ordinates of the row of broken lines in fig. 3.

EB1911 - Fourier's series - Fig. 3.png
Fig. 3.

The cosine series, which represents the same function for the interval 0 to l, may be found to be

1 l 2l ( cos 2πx + 1 cos 6πx + 1 cos 10πx + ... )
4 π2 l 32 l 52 l

This series represents for general values of x the ordinate of the set of broken lines in fig. 4.

EB1911 - Fourier's series - Fig. 4.png
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 ƒ(x), that function is in general represented by the series (3), consists in finding the sum of n+1 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 1/2 {ƒ(x + 0) + ƒ(x − 0)}, which is in general equal to ƒ(x). It will be convenient throughout to take −π to π as the given interval; any interval −l to l may be reduced to this by changing x into lx/π, and thus there is no loss of generality.

We find by an elementary process that

1/2 + cos (xx′) + cos 2(xx′) + . . . + cos n(xx′) = sin 2n + 1/2 (x′ – x) .
2 sin 1/2 (x′ – x)

Hence, with the new notation, the sum of the first n+1 terms of (3) is

1/π π π ƒ(x′) sin 2n + 1/2 (x′ – x) dx′.
2 sin 1/2 (x′ – x)

If we suppose ƒ(x) to be continued beyond the interval −π to π, in such a way that ƒ(x) = ƒ(x + 2π), we may replace the limits in this integral by x + π, xπ respectively; if we then put x′ − x = 2z, and let ƒ(x′) = F(z), the expression becomes , where ; this expression may be written in the form

(4)

We require therefore to find the limiting value, when m is indefinitely increased, of ; the form of the second integral being essentially the same. This integral, or rather the slightly more general one , when 0 < h1/2π, is known as Dirichlet’s integral. If we write , the integral becomes , 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 a < b, and suppose ƒ(x), φ(x) are two functions which have finite and determinate values everywhere in the interval except for a finite number of points; suppose further that the functions ƒ(x), φ(x) are integrable throughout the interval, and that as x increases from a to b the function ƒ(x) is monotone, i.e. either never diminishes or never increases; the theorem is that

when ξ is some point between a and b, and ƒ(a), ƒ(b) may be written for ƒ(a + 0), ƒ(b − 0) unless a or b is a point of discontinuity of the function ƒ(x).

To prove this theorem, we observe that, since the product of two integrable functions is an integrable function, b a ƒ(x) φ(x) dx exists, and may be regarded as the limit of the sum of a series

ƒ(x0) φ(x0) (x1x0) + ƒ(x1) φ(x1) (x2x1) + . . . + ƒ(xn−1) φ(xn−1) (xnxn−1)

where x0 = a, xn = b and x1, x2 . . . xn−1 are n − 1 intermediate points. We can express φ(xr) (xr+1xr) in the form Yr+1 − Yr, by putting

Yr = ΣK=r K=1 φ (xK−1) (xKxK−1), Y0 = 0.

Writing Xr for ƒ(xr), the series becomes

X0 (Y1 − Y0) + X1 (Y2 − Y1) + ... + Xn−1 (Yn − Yn−1)

or

Y1 (X0 − X1) + Y2 (X1 − X2) + ... + Yn(Xn−1 − Xn) + YnXn.

Now, by supposition, all the numbers Y1, Y2 ... Yn are finite, and all the numbers Xr−1 − Xr are of the same sign, hence by a known algebraical theorem the series is equal to M (X0 − Xn) + YnXn, where M is a number intermediate between the greatest and the least of the numbers Y1, Y2, ... Yn. This remains true however many partial intervals are taken, and therefore, when their number is increased indefinitely, and their breadths are diminished indefinitely according to any law, we have

when M is intermediate between the greatest and least values which can have, when x is in the given integral. Now this integral is a continuous function of its upper limit x, and therefore there is a value of x in the interval, for which it takes any particular value between the greatest and least values that it has. There is therefore a value φ between a and b, such that , hence

If the interval contains any finite numbers of points of discontinuity of ƒ(x) or φ(x), the method of proof still holds good, provided these points are avoided in making the subdivisions; in particular if either of the ends be a point of discontinuity of ƒ(x), we write ƒ(a + 0) or ƒ(b − 0), for ƒ(a) or ƒ(b), it being assumed that these limits exist.

Functions, with Limited Variation.—The condition that ƒ(x), in the mean-value theorem, either never increases or never diminishes as x increases from a to b, places a restriction upon the applications of the theorem. We can, however, show that a function ƒ(x) which is finite and continuous between a and b, except for a finite number of ordinary discontinuities, and which only changes from increasing to diminishing or vice versa, a finite number of times, as x increases from a to b, may be expressed as the difference of two functions ƒ1(x), ƒ2(x), neither of which ever diminishes as x passes from a to b, and that these functions are finite and continuous, except that one or both of them are discontinuous at the points where the given function is discontinuous. Let α, β be two consecutive points at which ƒ(x) is discontinuous, consider any point x1, such that αx1β, and suppose that at the points M1, M2 ... Mr between α and x1, ƒ(x) is a maximum, and at m1, m2 ... mr, it is a minimum; we will suppose, for example, that the ascending order of values is α, M1, m1, M2, m2 ... Mr, mr, x1; it will make no essential difference in the argument if m1 comes before M1, or if Mr immediately precedes x1, Mr−1 being then the last minimum.

Let

ψ(x1) = [ƒ(M1) − ƒ(α + 0)] + [ƒ(M2) − ƒ(m1)] + ... + [ƒ(Mr) − ƒ(mr−1)] + [ƒ(x1) − ƒ(mr)];

now let (x1) increase until it reaches the value (Mr+1) at which ƒ(x) is again a maximum, then let

ψ(x1) = [ƒ(M1) − ƒ(α + 0)] + [ƒ(M2) − ƒ(m1)] + . . . + [ƒ(Mr) − ƒ(mr−1)] + [ƒ(Mr+1) − ƒ(mr)];

and suppose as x increases beyond the value Mr+1, ψ(x1) remains constant until the next minimum mr+1 is reached, when it again becomes variable; we see that ψ(x1) is essentially positive and never diminishes as x increases.

Let

χ(x1) = [ƒ(M1) − f(m1)] + [ƒ(M2) − ƒ(m1)] + ... + [ƒ(Mr) − ƒ(mr)],

then let x1 increase until it is beyond the next maximum Mr+1, and then let

χ(x1) = [ƒ(M1) − ƒ(m1)] + [ƒ(M2) − ƒ(m1)] + . . . + [ƒ(Mr) − ƒ(mr)] + [ƒ(Mr+1) − ƒ(x1)]

thus χ(x1) never diminishes, and is alternately constant and variable. We see that ψ(x1) − χ(x1) is continuous as x1 increases from α to β, and that ψ(x1) − χ(x1) = ƒ(x1) − ƒ(α + 0), and when x1 reaches β, we have ψ(β) − χ(x1) = ƒ(β − 0) − ƒ(α + 0). Hence it is seen that between α and β, ƒ(x) = [ψ(x) + ƒ(α + 0)] − χ(x), where ψ(x) + ƒ(α + 0), χ(x) are continuous and never diminish as x increases; the same reasoning applies to every continuous portion of ƒ(x), for which the functions ψ(x), χ(x) are formed in the same manner; we now take ƒ1(x) = ψ(x) + ƒ(α + 0) + C, ƒ2(x) = χ(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 ƒ1(x) nor ƒ2(x) diminishes as x increases through a value for which ƒ(x) is discontinuous. We thus see that ƒ(x) = ƒ1(x) − ƒ2(x), where ƒ1(x), ƒ2(x) never diminish as x increases from a to b, and are discontinuous only where ƒ(x) is so. The function ƒ(x) is a particular case of a class of functions defined and discussed by Jordan, under the name “functions with limited variation” (fonctions à variation bornée); 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 ƒ(x) arbitrarily given between the values −π and +π, 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 0 to 1/2π, 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:

1. Since

this result holds however large the odd integer m may be.

2. If 0 < α < βπ/2,

β α sin mz dz = 1 γ α sin mz dz + 1 β γ sin mz dz
sin z sin α sin β

where α < γ < β, hence

|∫ β α sin mz dz | < 2 ( 1 + 1 ) < 4 ;
sin z m sin α sin β m sin α

a precisely similar proof shows that |∫ β α sin mz/z dz | < 4/mα,

hence the integrals β α sin mz/sin zdz, β α sin mz/zdz, converge to the limit zero, as m is indefinitely increased.

3. If α > 0, |∫ α sin θ/θ dθ | cannot exceed 1/2π. For by the mean-value theorem |∫ h α sin θ/θdθ | < 2/α + 2/h,

hence | Lh = ∞ h α sin θ/θ dθ | ≦ 2/α

in particular if απ |∫ α sin θ/θdθ |2/π < π/2.

Again d/dα α sin θ/θdθ = − sin α/α, α > 0,

therefore α sin θ/θdθ increases as α diminishes, when θ < α < π;

but lim α=0 α sin θ/θdθ = π/2, hence |∫ α sin θ/θ) dθ | < π/2,

where α < π, and < 2/π where απ. It follows that

|∫ β α sinθ dθ |π, provided 0 ≦ α < β.
θ

To find the limit of π/2 0 F(z) sin mz/sin zdz, we observe that it may be written in the form

F(0)   π/2 0 sin mz dz + μ 0 {F(z) − F(0)} sin mz dz + π/2 μ {F(z) − F(0)} sin mz dz
sin z sin z sin z

where μ is a fixed number as small as we please; hence if we use lemma (1), and apply the second mean-value theorem,

π/2 0 F(z) sin mz dz π F(0) = μ 0 {F(z) − F(0)} z   sin mz dz
sin z 2 sin z z
+ {F(μ + 0) − F(0)} ξ¹ μ sin mz dz + [F (1/2π − 0) − F(0)] π/2 ξ¹ sin mz dz
sin z sin z

when ξ¹ lies between μ and 1/2π. When m is indefinitely increased, the two last integrals have the limit zero in virtue of lemma (2). To evaluate the first integral on the right-hand side, let G(z) = {F(z) − F(0)} (z / sin z), and observe that G(z) increases as z increases from 0 to μ, hence if we apply the mean value theorem

|∫ μ 0 G(μ) sin mz dz | = | G(μ) μ ξ sin mz dz | = | G(μ) mμ mξ sin θ dθ | < πG(μ),
z z θ

where 0 < ξ < μ, since G(z) has the limit zero when z = 0. If ε be an arbitrarily chosen positive number, a fixed value of μ may be so chosen that πG(μ) < 1/2 ε, and thus that |∫ μ 0 G(z) sin mx/z dz | < 1/2 ε. When μ has been so fixed, m may now be so chosen that

|∫ 1/2π 0 F(z) sin mz dz π F(0) | < ε.
sin z 2

It has now been shown that when m is indefinitely increased π/2 0 F(z) sin mz/sin z dzπ/2 F(0) has the limit zero.

Returning to the form (4), we now see that the limiting value of

1 π/2 0 F(z) sin mz + 1 π/2 0 F(−z) sin mz dz is 1/2 {F(+0) + F(−0)};
π sin z π sin z

hence the sum of n + 1 terms of the series

1 l l ƒ(x) dx + 1 Σ l l ƒ(x¹) cos nπ(xx¹) dx
2l l l

converges to the value 1/2 {ƒ(x + 0) + ƒ(x − 0)}, or to ƒ(x) at a point where ƒ(x) is continuous, provided ƒ(x) satisfies Dirichlet’s conditions for the interval from −l to l.

Proof that Fourier’s Series is in General Uniformly Convergent.—To prove that Fourier’s Series converges uniformly to its sum for all values of x, provided that the immediate neighbourhoods of the points of discontinuity of ƒ(x) are excluded, we have

|∫ π/2 F(z) sin mz dz π F(0) | < πG(μ) + 4 {F(μ + 0) − F(0)} + 4 {F(1/2π − 0) − F(0)}
sin z 2 m sin μ m sin ξ¹
< πμ {ƒ(x + 2μ) − ƒ(x)} + 4 {ƒ(x + 2μ) − ƒ(x)} + 4 {ƒ(x + π) − ƒ(x)}.
sin μ m sin μ m sin ξ¹

Using this inequality and the corresponding one for F(−z), we have

|S2n+1(x) − ƒ(x)| < μ cosec μ [|ƒ(x + 2μ) − ƒ(x)| + |ƒ(x − 2μ) − ƒ(x)|] + A|m cosec μ,

where A is some fixed number independent of m. In any interval (a, b) in which ƒ(x) is continuous, a value μ1 of μ can be chosen such that, for every value of x in (a, b), |ƒ(x + 2μ) − ƒ(x)|, |ƒ(x − 2μ) − ƒ(x)| are less than an arbitrarily prescribed positive number ε, provided μ = μ1. Also a value μ2 of μ can be so chosen that εμ2 cosec μ2 < 1/2 η, where η is an arbitrarily assigned positive number. Take for μ the lesser of the numbers μ1, μ2, then |S2n+1 − ƒ(x)| < η + A|m cosec μ for every value of x in (a, b). It follows that, since η and m are independent of x, |S2n+1 − ƒ(x)| < 2ε, provided n is greater than some fixed value n1 dependent only on ε. Therefore S2n+1 converges to ƒ(x) uniformly in the interval (a, b).

Case of a Function with Infinities.—The limitation that ƒ(x) must be numerically less than a fixed positive number throughout the interval may, under a certain restriction, be removed. Suppose F(z) is indefinitely great in the neighbourhood of the point z = c, and is such that the limits of the two integrals c±ε c F(z) dz are both zero, as ε is indefinitely diminished, then

π/2 0 F(z) sin mz/sin zdz denotes the limit when ε = 0, ε¹ = 0 of cε 0 F(z) sin mz/sin zdz + π/2 c+ε¹ F(z) sin mz/sin zdz, both these limits existing; the first of these integrals has 1/2πF(+0) for its limiting value when m is indefinitely increased, and the second has zero for its limit. The theorem therefore holds if F(z) has an infinity up to which it is absolutely integrable; this will, for example, be the case if F(z) near the point C is of the form x(z)(zc)μ + ψ(z), where χ(c), ψ(c) are finite, and 0 < μ < 1. It is thus seen that ƒ(x) 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 ƒ(x) is everywhere finite within the given interval −π to +π, it can be shown that an, bn, the coefficients of cos nx, sin nx in the series which represent the function, are such that nan, nbn, however great n is, are each less than a fixed finite quantity. For writing ƒ(x) = ƒ1(x) − ƒ2(x), we have

π π ƒ1(x) cos nxdx = ƒ1(−π + 0) ξ π cos nxdx + ƒ1(π − 0) π ξ cos nxdx

hence

π π ƒ1(x) cos nxdx = ƒ1(−π + 0) sin nξ + ƒ1(π − 0) sin nξ
n n

with a similar expression, with ƒ2(x) for ƒ1(x), ξ being between π and −π; the result then follows at once, and is obtained similarly for the other coefficient.

If ƒ(x) is infinite at x = c, and is of the form φ(x) / (xc)K near the point c, where 0 < K < 1, the integral π π ƒ(x)cos nxdx contains portions of the form ε+ε c φ(x)/xcK cos nxdx c cε φ(x)/xcK cos nxdx; consider the first of these, and put x = c + u, it thus becomes ε 0 φ(c + u)/uK cos n(c + u)du, which is of the form

φ(c + θε) ε 0 cos n(c + u)/uKdu; now let nu = v, the integral becomes

φ(c + θε) { cos nc nε 0  cos v dv sin nc nε 0  sin v dv };
n1−K vK n1−K vK

hence n1−K π π ƒ(x) cos nxdx becomes, as n is definitely increased, of the form

φ(c) { cos nc 0 cos v dv − sin nc 0 sin v dv }
vK vK

which is finite, both the integrals being convergent and of known value. The other integral has a similar property, and we infer that n1−K an, n1−K bn are less than fixed finite numbers.

The Differentiation of Fourier’s Series.—If we assume that the differential coefficient of a function ƒ(x) represented by a Fourier’s Series exists, that function ƒ′(x) is not necessarily representable by the series obtained by differentiating the terms of the Fourier’s Series, such derived series being in fact not necessarily convergent. Stokes has obtained general formulae for finding the series which represent ƒ′(x), ƒ″(x)—the successive differential coefficients of a limited function ƒ(x). As an example of such formulae, consider the sine series (1); ƒ(x) is represented by

2 Σ sin nπx l 0 ƒ(x) sin nπx dx;
l l l

on integration by parts we have

l 0 ƒ(x) sin nπx dx = l [ ƒ(+0) ± ƒ(l − 0) + Σ cos nπa {ƒ(α + 0) − ƒ(α − 0)} ]
l nπ l
+ l l 0 ƒ′(x) cos nπx dx
nπ l

where α represent the points where ƒ(x) is discontinuous. Hence if f(x) is represented by the series Σan sin nπx/l, and ƒ′(x) by the series Σbn cos nπx/l, we have the relation

bn = nπ an 2 [ ƒ(+0) = ƒ(l − 0) + Σ cos nπα {ƒ(α + 0) − ƒ(α − 0)} ]
l l l

hence only when the function is everywhere continuous, and ƒ(+0) ƒ(l − 0) are both zero, is the series which represents ƒ′(x) obtained at once by differentiating that which represents ƒ(x). The form of the coefficient an discloses the discontinuities of the function and of its differential coefficients, for on continuing the integration by parts we find

αn = 2 [ ƒ(+0) = ƒ(l − 0) + Σ cos nπα {ƒ(α + 0) − ƒ(α − 0)} ]
nπ l
+ 2l [ ƒ′(+0) = ƒ′(l − 0) + Σ cos nπβ {ƒ′(β + 0) − ƒ′(β − 0)} ] + &c.
n2π2 l

where β are the points at which ƒ′(x) is discontinuous.

History and Literature of the theory

The history of the theory of the representation of functions by series of sines and cosines is of great interest in connexion with the progressive development of the notion of an arbitrary function of a real variable, and of the peculiarities which such a function may possess; the modern views on the foundations of the infinitesimal calculus have been to a very considerable extent formed in this connexion (see Function). The representation of functions by these series was first considered in the 18th century, in connexion with the problem of a vibrating cord, and led to a controversy as to the possibility of such expansions. In a memoir published in 1747 (Memoirs of the Academy of Berlin, vol. iii.) D’Alembert showed that the ordinate y at any time t of a vibrating cord satisfies a differential equation of the form δ2y/δt2 = a2δ2y/δx2, where x is measured along the undisturbed length of the cord, and that with the ends of the cord of length l fixed, the appropriate solution is y = ƒ(at + x) − ƒ(atx), where ƒ is a function such that ƒ(x) = ƒ(x + 2l); in another memoir in the same volume he seeks for functions which satisfy this condition. In the year 1748 (Berlin Memoirs, vol. iv.) Euler, in discussing the problem, gave ƒ(x) = α sin πx/l + β sin 2πx/l + . . . as a particular solution, and maintained that every curve, whether regular or irregular, must be representable in this form. This was objected to by D’Alembert (1750) and also by Lagrange on the ground that irregular curves are inadmissible. D. Bernoulli (Berlin Memoirs, vol. ix., 1753) based a similar result to that of Euler on physical intuition; his method was criticized by Euler (1753). The question was then considered from a new point of view by Lagrange, in a memoir on the nature and propagation of sound (Miscellanea Taurensia, 1759; Œuvres, vol. i.), who, while criticizing Euler’s method, considers a finite number of vibrating particles, and then makes the number of them infinite; he did not, however, quite fully carry out the determination of the coefficients in Bernoulli’s Series. These mathematicians were hampered by the narrow conception of a function, in which it is regarded as necessarily continuous; a discontinuous function was considered only as a succession of several different functions. Thus the possibility of the expansion of a broken function was not generally admitted. The first cases in which rational functions are expressed in sines and cosines were given by Euler (Subsidium calculi sinuum, Novi Comm. Petrop., vol. v., 1754–1755), who obtained the formulae

1/2 φ = sin φ1/2 sin 2φ + 1/3 sin 3φ ...
π2 φ2 = cos φ1/4 cos 2φ + 1/9 cos 3φ ...
12 4

In a memoir presented to the Academy of St Petersburg in 1777, but not published until 1798, Euler gave the method afterwards used by Fourier, of determining the coefficients in the expansions; he remarked that if Φ is expansible in the form

A + B cos φ + C cos 2φ + ..., then A = 1 π 0 Φ dφ, B = 2 π 0 Φ cos φ dφ, &c.
π π

The second period in the development of the theory commenced in 1807, when Fourier communicated his first memoir on the Theory of Heat to the French Academy. His exposition of the present theory is contained in a memoir sent to the Academy in 1811, of which his great treatise the Théorie analytique de la chaleur, published in 1822, is, in the main, a reproduction. Fourier set himself to consider the representation of a function given graphically, and was the first fully to grasp the idea that a single function may consist of detached portions given arbitrarily by a graph. He had an accurate conception of the convergence of a series, and although he did not give a formally complete proof that a function with discontinuities is representable by the series, he indicated in particular cases the method of procedure afterwards carried out by Dirichlet. As an exposition of principles, Fourier’s work is still worthy of careful perusal by all students of the subject. Poisson’s treatment of the subject, which has been adopted in English works (see the Journal de l’école polytechnique, vol. xi., 1820, and vol. xii., 1823, and also his treatise, Théorie de la chaleur, 1835), depends upon the equality

π π ƒ(α) 1h2 dα = 1 π π ƒ(α) dα + 1 Σ hn π π ƒ(α) cos n(xα) dα
1 − 2h cos (xα) + h2 2π π

where 0 < h < 1; the limit of the integral on the left-hand side is evaluated when h = 1, and found to be 1/2 {ƒ(x + 0) + ƒ(x − 0)}, the series on the right-hand side becoming Fourier’s Series. The equality of the two limits is then inferred. If the series is assumed to be convergent when h = 1, by a theorem of Abel’s its sum is continuous with the sum for values of h less than unity, but a proof of the convergency for h = 1 is requisite for the validity of Poisson’s proof; as Poisson gave no such proof of convergency, his proof of the general theorem cannot be accepted. The deficiency cannot be removed except by a process of the same nature as that afterwards applied by Dirichlet. The definite integral has been carefully studied by Schwarz (see two memoirs in his collected works on the integration of the equation δ2u/δx2 + δ2u/δy2 = 0), who showed that the limiting value of the integral depends upon the manner in which the limit is approached. Investigations of Fourier’s Series were also given by Cauchy (see his “Mémoire sur les développements des fonctions en séries périodiques,” Mém. de l’Inst., vol. vi., also Œuvres complètes, vol. vii.); his method, which depends upon a use of complex variables, was accepted, with some modification, as valid by Riemann, but one at least of his proofs is no longer regarded as satisfactory. The first completely satisfactory investigation is due to Dirichlet; his first memoir appeared in Crelle’s Journal for 1829, and the second, which is a model of clearness, in Dove’s Repertorium der Physik. Dirichlet laid down certain definite sufficient conditions in regard to the nature of a function which is expansible, and found under these conditions the limiting value of the sum of n terms of the series. Dirichlet’s determination of the sum of the series at a point of discontinuity has been criticized by Schläfli (see Crelle’s Journal, vol. lxxii.) and by Du Bois-Reymond (Mathem. Annalen, vol. vii.), who maintained that the sum is really indeterminate. Their objection appears, however, to rest upon a misapprehension as to the meaning of the sum of the series; if x1 be the point of discontinuity, it is possible to make x approach x1, and n become indefinitely great, so that the sum of the series takes any assigned value in a certain interval, whereas we ought to make x = x1 first and afterwards n = ∞, and no other way of going to the double limit is really admissible. Other papers by Dircksen (Crelle, vol. iv.) and Bessel (Astronomische Nachrichten, vol. xvi.), on similar lines to those by Dirichlet, are of inferior importance. Many of the investigations subsequent to Dirichlet’s have the object of freeing a function from some of the restrictions which were imposed upon it in Dirichlet’s proof, but no complete set of necessary and sufficient conditions as to the nature of the function has been obtained. Lipschitz (“De explicatione per series trigonometricas,” Crelle’s Journal, vol. lxiii., 1864) showed that, under a certain condition, a function which has an infinite number of maxima and minima in the neighbourhood of a point is still expansible; his condition is that at the point of discontinuity β, |ƒ(β + δ) − f(β)| < Bδα as δ converges to zero, B being a constant, and α a positive exponent. A somewhat wider condition is

ƒ(β + δ) − ƒ(β)} log δ = 0,
δ = 0

for which Lipschitz’s results would hold. This last condition is adopted by Dini in his treatise (Sopra la serie di Fourier, &c., Pisa, 1880).

The modern period in the theory was inaugurated by the publication by Riemann in 1867 of his very important memoir, written in 1854, Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe. The first part of his memoir contains a historical account of the work of previous investigators; in the second part there is a discussion of the foundations of the Integral Calculus, and the third part is mainly devoted to a discussion of what can be inferred as to the nature of a function respecting the changes in its value for a continuous change in the variable, if the function is capable of representation by a trigonometrical series. Dirichlet and probably Riemann thought that all continuous functions were everywhere representable by the series; this view was refuted by Du Bois-Reymond (Abh. der Bayer. Akad. vol. xii. 2). It was shown by Riemann that the convergence or non-convergence of the series at a particular point x depends only upon the nature of the function in an arbitrarily small neighbourhood of the point x. The first to call attention to the importance of the theory of uniform convergence of series in connexion with Fourier’s Series was Stokes, in his memoir “On the Critical Values of the Sums of Periodic Series” (Camb. Phil. Trans., 1847; Collected Papers, vol. i.). As the method of determining the coefficients in a trigonometrical series is invalid unless the series converges in general uniformly, the question arose whether series with coefficients other than those of Fourier exist which represent arbitrary functions. Heine showed (Crelle’s Journal, vol. lxxi., 1870, and in his treatise Kugelfunctionen, vol. i.) that Fourier’s Series is in general uniformly convergent, and that if there is a uniformly convergent series which represents a function, it is the only one of the kind. G. Cantor then showed (Crelle’s Journal, vols. lxxii. lxxiii.) that even if uniform convergence be not demanded, there can be but one convergent expansion for a function, and that it is that of Fourier. In the Math. Ann. vol. v., Cantor extended his investigation to functions having an infinite number of discontinuities. Important contributions to the theory of the series have been published by Du Bois-Reymond (Abh. der Bayer. Akademie, vol. xii., 1875, two memoirs, also in Crelle’s Journal, vols. lxxiv. lxxvi. lxxix.), by Kronecker (Berliner Berichte, 1885), by O. Hölder (Berliner Berichte, 1885), by Jordan (Comptes rendus, 1881, vol. xcii.), by Ascoli (Math. Annal., 1873, and Annali di matematica, vol. vi.), and by Genocchi (Atti della R. Acc. di Torino, vol. x., 1875). Hamilton’s memoir on “Fluctuating Functions” (Trans. R.I.A., vol. xix., 1842) may also be studied with profit in this connexion. A memoir by Brodén (Math. Annalen, vol. lii.) contains a good investigation of some of the most recent results on the subject. The scope of Fourier’s Series has been extended by Lebesgue, who introduced a conception of integration wider than that due to Riemann. Lebesgue’s work on Fourier’s Series will be found in his treatise, Leçons sur les séries trigonométriques (1906); also in a memoir, “Sur les séries trigonométriques,” Annales sc. de l’école normale supérieure, series ii. vol. xx. (1903), and in a paper “Sur la convergence des séries de Fourier,” Math. Annalen, vol. lxiv. (1905).

Authorities.—The foregoing historical account has been mainly drawn from A. Sachse’s work, “Versuch einer Geschichte der Darstellung willkürlicher Functionen einer Variabeln durch trigonometrische Reihen,” published in Schlömilch’s Zeitschrift für Mathematik, Supp., vol. xxv. 1880, and from a paper by G. A. Gibson “On the History of the Fourier Series” (Proc. Ed. Math. Soc. vol. xi.). Reiff’s Geschichte der unendlichen Reihen may also be consulted, and also the first part of Riemann’s memoir referred to above. Besides Dini’s treatise already referred to, there is a lucid treatment of the subject from an elementary point of view in C. Neumann’s treatise, Über die nach Kreis-, Kugel- und Cylinder-Functionen fortschreitenden Entwickelungen. Jordan’s discussion of the subject in his Cours d’analyse is worthy of attention: an account of functions with limited variation is given in vol. i.; see also a paper by Study in the Math. Annalen, vol. xlvii. On the second mean-value theorem papers by Bonnet (Brux. Mémoires, vol. xxiii., 1849, Lionville’s Journal, vol. xiv., 1849), by Du Bois-Reymond (Crelle’s Journal, vol. lxxix., 1875), by Hankel (Zeitschrift für Math. und Physik, vol. xiv., 1869), by Meyer (Math. Ann., vol. vi., 1872) and by Hölder (Göttinger Anzeigen, 1894) may be consulted; the most general form of the theorem has been given by Hobson (Proc. London Math. Soc., Series II. vol. vii., 1909). On the theory of uniform convergence of series, a memoir by W. F. Osgood (Amer. Journal of Math. xix.) may be with advantage consulted. On the theory of series in general, in relation to the functions which they can represent, a memoir by Baire (Annali di matematica, Series III. vol. iii.) is of great importance. Bromwich’s Theory of Infinite Series (1908) contains much information on the general theory of series. Bôcher’s “Introduction to the Theory of Fourier’s Series,” Annals of Math., Series II. vol. vii., 1906, will be found useful. See also Carslaw’s Introduction to the Theory of Fourier’s Series and Integrals, and the Mathematical Theory of the Conduction of Heat (1906). A full account of the theory will be found in Hobson’s treatise On the Theory of Functions of a Real Variable and on the Theory of Fourier’s Series (1907).  (E. W. H.)