4c / [pi]x 1 3[pi]x 1 5[pi]x \
---- ( cos ----- - -- cos ------ + -- cos ------ - + ... )
[pi] \ l 3 l 5 l /
during the interval 0 to l; for general values of x the series
represents the ordinate of the broken line in fig. 2, except that it
vanishes at the points (1/2)l, (3/2)l....
[Illustration: FIG. 2.]
(b) Let [f](x) = x from 0 to 1/2 l, and f(x) = l - x, from 1/2 l to l;
then
_ _ _
/ l n[pi]x / 1/2 l n[pi]x / l n[pi]x
| [f](x)sin ------ dx= | x sin ------ dx + | (l - x)sin ------ dx
_/ 0 l _/ 0 l _/1/2 l l
l^2 n[pi] l^2 n[pi] l^2n / n[pi] \
= - ------ cos ----- + --------- sin ----- + ----- (cos ----- - cos n[pi] )
2n[pi] 2 n^2[pi]^2 2 n[pi] \ 2 /
l^2 l^2 n[pi] l^2 n[pi] 2l^2 n[pi]
+ ----- cos n[pi] - ------ cos ----- + --------- sin ----- = --------- sin -----
n[pi] 2n[pi] 2 n^2[pi]^2 2 n^2[pi]^2 2
hence the sine series is
4l / nx 1 3[pi]x 1 5[pi]x \
------ (sin -- - --- sin ------ + --- sin ------ - ... )
[pi]^2 \ l 3^2 l 5^2 l /
For general values of x, the series represents the ordinates of the
row of broken lines in fig. 3.
[Illustration: FIG. 3.]
The cosine series, which represents the same function for the interval
0 to l, may be found to be
1 2l / 2[pi]x 1 6[pi]x 1 10[pi]x \
-- l - ------ (cos ------ + --- cos ------ + --- cos ------- + ... )
4 [pi]^2 \ l 3^2 l 5^2 l /
This series represents for general values of x the ordinate of the set
of broken lines in fig. 4.
[Illustration: FIG. 4.]
_Dirichlet's Integral._--The method indicated by Fourier, but first
carried out rigorously by Dirichlet, of proving that, with certain
restrictions as to the nature of the function [f](x), that function is
in general represented by the series (3), consists in finding the sum
of n+1 terms of that series, and then investigating the limiting value
of the sum, when n is increased indefinitely. It thus appears that the
series is convergent, and that the value towards which its sum
converges is 1/2 {[f](x + 0) + [f](x - 0)}, which is in general equal
to [f](x). It will be convenient throughout to take -[pi] to [pi] as
the given interval; any interval -l to l may be reduced to this by
changing x into lx/[pi], and thus there is no loss of generality.