are equivalent to
| − ∫ −l 0 ƒ(x) cos | nπx | dx, + ∫ −l 0 ƒ(x) sin | nπx | dx, |
| l | l |
thus the series is
| 1 | ∫ l −l ƒ(x) dx + | 1 | Σ∞ 1 cos | nπx | ∫ l −l ƒ(x) cos | nπx | dx + | 1 | Σ∞ 1 sin | nπx | ∫ l −l ƒ(x) sin | nπx | dx, |
| 2l | l | l | l | l | l | l |
which may be written
| 1 | ∫ l −l ƒ(x′) dx′ + | 1 | Σ∞ 1 ∫ l −l ƒ(x′) cos | nπ (x − x′) | dx′. |
| 2l | l | l |
(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 ½ {ƒ(x + 0) + ƒ(x − 0)}, 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 ½ {ƒ(+0) + ƒ(−0)}; 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 ½ {ƒ(l) + ƒ(−l)}.
Examples of Fourier’s Series.—(a) Let ƒ(x) be given from 0 to l, by ƒ(x) = c, when 0 ≦ x < ½ l, and by f(x)= −c from ½ 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