Fourier sine coefficient ($b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\sin (n\pi x/L)dx$)

For an integrable function on $[-L,L]$, with $L>0$ and integer $n\ge 1$, 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 $b_n=(2/L){\int }_0^{L}f(x)\sin (n\pi x/L)dx$. These are also the coefficients of the Fourier sine series of the restriction to $(0,L)$.