Definition:Half-Range Fourier Cosine Series

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

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

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

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

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

Also see