Definition:Half-Range Fourier Sine Series/Formulation 2

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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:

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

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

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


Also known as

Some sources give the half-range Fourier series as Fourier's half-range series.

Some sources give them as just the half-range series.


Also see


Sources