Definition:Half-Range Fourier Cosine Series/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 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 $n \in \Z_{\ge 0}$:
- $a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$
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 Even Function over Symmetric Range, which justifies the definition
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cosine series: 2.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Fourier's half-range series
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cosine series: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Fourier's half-range series