Definition:Half-Range Fourier Sine Series/Formulation 2
Jump to navigation
Jump to search
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
- Fourier Series for Odd Function over Symmetric Range, which justifies the definition
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 7$. Fourier Series over a General Range $\tuple {a, b}$: $(4)$