Definition:Half-Range Fourier Series/Range Pi

Definition

Half-Range Fourier Cosine Series

Let $\map f x$ be a real function defined on the interval $\openint 0 \pi$.

Then the half-range Fourier cosine series of $\map f x$ over $\openint 0 \pi$ is the series:

$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty a_n \cos n x$

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

$a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$

Half-Range Fourier Sine Series

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 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

• Definition:Half-Range Fourier Series

• Results about half-range Fourier series can be found here.

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): half-range series
• 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): half-range series