# Definition:Half-Range Fourier Sine Series

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## Definition

### Formulation 1

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

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

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

where for all $n \in \Z_{> 0}$:

$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$

### Formulation 2

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 $n \in \Z_{> 0}$:

$B_m = \displaystyle \frac 2 {b - a} \int_a^b \map f x \sin\frac {m \pi \paren {x - a} } {b - a} \rd x$

## Half-Range Fourier Sine Series on Range of $\pi$

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:

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

where for all $n \in \Z_{> 0}$:

$b_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$