Definition:Half-Range Fourier Sine Series/Formulation 2

Definition
Let $\map f x$ be a real function defined on the interval $\openint a b$.

Then the  half-range Fourier sine series of $\map f x$ over $\openint a b$ is the series:


 * $\displaystyle \map f x \sim \sum_{m \mathop = 1}^\infty B_m \sin \frac {m \pi \paren {x - a} } {b - a}$

where for all $n \in \Z_{> 0}$:
 * $B_m = \displaystyle \frac 2 {b - a} \int_a^b \map f x \sin\frac {m \pi \paren {x - a} } {b - a} \rd x$