Definition:Half-Range Fourier Cosine Series/Formulation 2

Definition
Let $f \left({x}\right)$ be a real function defined on the interval $\left[{a \,.\,.\, b}\right]$.

Then the  half-range Fourier cosine series of $f \left({x}\right)$ over $\left[{a \,.\,.\, b}\right]$ is the series:


 * $\displaystyle f \left({x}\right) \sim \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty A_m \cos \frac {m \pi \left({x - a}\right)} {b - a}$

where for all $m \in \Z_{\ge 0}$:
 * $A_m = \displaystyle \frac 2 {b - a} \int_a^b f \left({x}\right) \cos \frac {m \pi \left({x - a}\right)} {b - a} \, \mathrm d x$