\begin{definition}[Definition:Half-Range Fourier Sine Series/Range Pi]
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:
:$\map f x \sim \ds \sum_{n \mathop = 1}^\infty b_n \sin n x$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
\end{definition}