| ∫ l 0 ƒ(x) sin | nπx | dx = c ∫ ½ l 0 sin | nπx | dx − c ∫ l ½l sin | nπx | dx |
| l | l | l |
| = | cl | (cos nπ − 2 cos ½nπ + 1). |
| nπ |
This vanishes if n is odd, and if n = 4m, but if n = 4m + 2 it is equal to 4cl / nπ; the series is therefore
| 4c | ( | l | sin | 2πx | + | 1 | sin | 6πx | + | 1 | sin | 10πx | + ... ). |
| π | 2 | l | 3 | l | 5 | l |
For unrestricted values of x, this series represents the ordinates of the series of straight lines in fig. 1, except that it vanishes at the points 0, ½ l, l, 3⁄2 l ...
We find similarly that the same function is represented by the series
| 4c | ( cos | πx | − | 1 | cos | 3πx | + | 1 | cos | 5πx | − + ... ) |
| π | 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 ½ l, 3⁄2 l ...