Definition:Half-Range Fourier Series

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Definition

Half-Range Fourier Cosine Series

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


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

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

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

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


Half-Range Fourier Sine Series

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


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

$\map f x \sim \ds \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$

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

$b_n = \ds \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$


Half-Range Fourier Series on Range of $\pi$

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

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


Source of Name

This entry was named for Joseph Fourier.


Sources

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