hence ƒ(x) is represented for the interval 0 to l by the series of cosines
| 1 | ∫ l 0 ƒ(x) dx + | 2 | Σ∞ 1 cos | nπx | ∫ l 0 ƒ(x) cos | nπx | dx. |
| l | l | l | l |
(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 o 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 ½ {ƒ(x) − ƒ(−x)} which is an odd function, and thus this function is represented for the interval −l to +l by
| 2 | Σ sin | nπx | ∫ l 0 ½ {ƒ(x) − ƒ(−x)} sin | nπx | dx; |
| l | l | l |
we can also express ½ {ƒ(x) + ƒ(−x)}, which is an even function, by means of (2), thus for the interval −l to +l this function is represented by
| 1 | ∫ l 0 ½ {ƒ(x) + ƒ(−x)} dx + | 2 | Σ∞ 1 cos | nπx | ∫ l 0 ½ {ƒ(x) + ƒ(−x)} cos | nπx | dx. |
| l | l | l | l |
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
| ∫ l 0 ƒ(−x) cos | nπx | dx, ∫ l 0 ƒ(−x) sin | nπx | dx |
| l | l |