Half-integer cosine expansion of a linear profile ($L-x=\frac{8L}{{\pi }^2}{\sum }_{n\ge 0}\frac{\cos ((2n+1)\pi x/(2L))}{(2n+1{)}^2}$)

For $L>0$ and $0\le x\le L$, 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 givesThus 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.