Fourier cosine coefficient ($a_n=\frac{2}{L}{\int }_0^{L}f(x)\cos (n\pi x/L)dx$)

On $[0,L]$, the Fourier cosine series convention $f(x)=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (n\pi x/L)$ uses the displayed coefficients, including $n=0$. Orthogonality of the cosine modes gives these formulas for square-integrable functions; the expansion is interpreted in the L2 norm until stronger convergence is established.