| φ(c + θε) { | cos nc | ∫ θε 0 | cos v | dv − | sin nc | ∫ θε 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