Definition:Half-Range Fourier Sine Series

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

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


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

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

Also see

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