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) / (x − c)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)x − cK cos nxdx ∫ c c−ε φ(x)x − cK 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 nπxl, and ƒ′(x) by the series Σbn cos nπxl, 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 δ2yδt2 = a2δ2yδx2, 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(at − x), 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 πxl + β sin 2πxl + . . . 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
| π2 | − | φ2 | = cos φ − 14 cos 2φ + 19 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) | 1 − h2 | dα = | 1 | ∫ π −π ƒ(α) dα + | 1 | Σ hn ∫ π −π ƒ(α) cos n(x − α) dα |
| 1 − 2h cos (x − α) + 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 12 {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 δ2uδx2 + δ2uδy2 = 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
