Definition:Half-Range Fourier Sine Series

Definition
Let $f \left({x}\right)$ be a real function defined on the interval $\left[{0 \,.\,.\, l}\right]$.

Then the  half-range Fourier sine series of $f \left({x}\right)$ over $\left[{0 \,.\,.\, l}\right]$ is the series:


 * $\displaystyle f \left({x}\right) \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} l$

where for all $n \in \Z_{> 0}$:
 * $b_n = \displaystyle \frac 2 l \int_0^l f \left({x}\right) \sin \frac {n \pi x} l \, \mathrm d x$

Also see

 * Fourier Series for Odd Function over Symmetric Range, which justifies the definition