Fourier Cosine Series

Definition

$f(x)\sim \frac{a_0}{2}+\sum _{n=1}^{\infty }{a}_n\cos (nx)$

Given $f\in C_p(0,\pi )$^[Piecewise Continuous],
an infinite series $\frac{a_0}{2}+\sum _{n=1}^{\infty }{a}_n\cos (nx),a_n=\frac{2}{\pi }{\int }_0^{\pi }f(x)\cos (nx)dx,{}_{(n=0,1,2)}$ is called the Fourier cosine series representation for $f$.