Definition:Half-Range Fourier Sine Series/Range Pi
< Definition:Half-Range Fourier Sine Series(Redirected from Definition:Half-Range Fourier Sine Series over Range Pi)
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Definition
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:
- $\map f x \sim \ds \sum_{n \mathop = 1}^\infty b_n \sin n x$
where for all $n \in \Z_{> 0}$:
- $b_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
Also see
- Fourier Series for Even Function over Symmetric Range, which justifies the definition
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Fourier's half-range series
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): sine series: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Fourier's half-range series
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): sine series: 2.