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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),
\end{displaymath} (1)

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

for $n=1$, 2, 3, .... The last Equality is true because
$\displaystyle f(x)\sin(nx)$ $\textstyle =$ $\displaystyle [-f(-x)][-\sin(-nx)]$  
  $\textstyle =$ $\displaystyle f(-x)\sin(-nx).$ (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.
\end{displaymath} (4)

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

© 1996-9 Eric W. Weisstein