Fourier Sine Series

If $f(x)$ is an Odd Function, then $a_n = 0$ and the Fourier Series collapses to

$$f(x) = \sum_{n=1}^\infty b_n\sin(nx),$$ (1)

where

$$b_n = {1\over\pi}\int^\pi_{-\pi} f(x)\sin(nx)\,dx = {2\over \pi}\int^\pi_0 f(x)\sin(nx)\,dx$$ (2)

for $n=1$, 2, 3, .... The last Equality is true because

$$f(x)\sin(nx) = [-f(-x)][-\sin(-nx)]$$

$$= f(-x)\sin(-nx).$$ (3)

Letting the range go to $L$,

$$b_n={2\over L} \int^L_0 f(x)\sin\left({n\pi x\over L}\right)\,dx.$$ (4)

See also Fourier Cosine Series, Fourier Series, Fourier Sine Transform