Definition:Half-Range Fourier Sine Series/Range Pi

Definition
Let $\map f x$ be a real function defined on the interval $\openint 0 \pi$.

Then the  half-range Fourier sine series of $\map f x$ over $\openint 0 \pi$ is the series:


 * $\map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin n x$

where for all $n \in \Z_{> 0}$:
 * $b_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$