Category:Definitions/Half-Range Fourier Series

This category contains definitions related to Half-Range Fourier Series.
Related results can be found in Category:Half-Range Fourier Series.

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 $n \in \Z_{> 0}$:

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