= Half-integer cosine expansion of a linear profile
{title2=$L-x=\frac{8L}{\pi^2}\sum_{n\geq0}\frac{\cos((2n+1)\pi x/(2L))}{(2n+1)^2}$}
For $L>0$ and $0\leq x\leq L$, set $\mu_n=(2n+1)\pi/(2L)$. The <cosine> modes satisfy a homogeneous <Neumann boundary condition> at zero and a homogeneous <Dirichlet boundary condition> at $L$. Their squared <norm> is $L/2$, and <integration by parts> gives
$$
\int_0^L(L-x)\cos(\mu_nx)\,dx=\frac{1-\cos(\mu_nL)}{\mu_n^2}=\frac1{\mu_n^2}.
$$
Thus their <Fourier coefficients> are $2/(L\mu_n^2)$, proving the displayed formula by <Sturm-Liouville theory>. One can also obtain completeness by extending evenly across zero and oddly across $L$: the resulting continuous $4L$-<periodic extension> has only odd-index <cosine> modes. The $n^{-2}$ coefficient bound proves <uniform convergence>, including both endpoints. This expansion supplies the initial transient for a <heat equation> with mixed endpoint conditions.
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