= Fourier sine coefficient
{c}
{title2=$b_n=\frac1L\int_{-L}^Lf(x)\sin(n\pi x/L)dx$}
For an integrable <function> on $[-L,L]$, with $L>0$ and <integer> $n\geq1$, the displayed <Fourier coefficient> multiplies the mode $\sin(n\pi x/L)$ in its real <Fourier series>. <Fourier orthogonality> fixes the normalization because the squared mode has <integral> $L$. If $f$ is <odd>, the product with the sine is <even>, so $b_n=(2/L)\int_0^Lf(x)\sin(n\pi x/L)dx$. These are also the coefficients of the <Fourier sine series> of the restriction to $(0,L)$.
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