Definition:Half-Range Fourier Sine Series/Range Pi

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

• Definition:Half-Range Fourier Sine Series

• Definition:Half-Range Fourier Cosine Series over Range Pi

• Definition:Half-Range Fourier Series over Range Pi
• Definition:Half-Range Fourier Series

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.