Definition:Half-Range Fourier Sine Series/Formulation 1
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Definition
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:
- $\map f x \sim \ds \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
- $b_n = \ds \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$
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
- Fourier Series for Odd Function over Symmetric Range, which justifies the definition
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 5$. Half-Range Sine Series: $(2)$
- 1968: Peter D. Robinson: Fourier and Laplace Transforms ... (previous) ... (next): $\S 1.3$. Fourier Series and Finite Fourier Transforms