Definition:Half-Range Fourier Cosine Series/Range Pi

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 $n \in \Z_{\ge 0}$:
 * $a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$