Definition:Half-Range Fourier Sine Series/Formulation 2

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

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


 * $\displaystyle f \left({x}\right) \sim \sum_{m \mathop = 1}^\infty B_m \sin \frac {m \pi \left({x - a}\right)} {b - a}$

where for all $n \in \Z_{> 0}$:
 * $B_m = \displaystyle \frac 2 {b - a} \int_a^b f \left({x}\right) \sin\frac {m \pi \left({x - a}\right)} {b - a} \, \mathrm d x$