| 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 = a² (δ²y / 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(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 (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
½ φ = sin φ − ½ sin 2φ + 1⁄3 sin 3φ ...
| π² | − | φ² | = cos φ − ¼ 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