Fourier Series

than in a formula. Periodicity can play strange tricks!Figure 1.2.1. The periodic function fx x.It is clear from figure 1.2.1 that the periodic function possesses discontinuities atx ,3,5,...To obtain the Fourier coefficients, use (1.2.2) and (1.2.4):a 0 1 −xdx 1 2x 2 2 − 0a m 1 x cosmx dx −1 1m cosmx 2 −1 xm sinmx −−1 − 1m 2 −1m − −1 m 01msinmx dxb m 1 −x sinmx dx 1 − x m cosmx −1 −1mcosmx dx 1 − m −1m −− m −1m 2 m −1m1where the result cosm −1 m has been used on a number of occasions in the derivation.Thus the final result isx~2 ∑n1−1n1nsinnx 2 sinx − 1 2 sin2x 1 3 sin3x − 1 4sin4x ... .The fact that the resulting series contains only sine terms is not surprising, since fx x isan odd function and so the series should only be composed of terms with the same property.14