= Quadratic arch expansion in half-integer cosine modes
{title2=$b^2-y^2=\frac{32b^2}{\pi^3}\sum_{n\geq0}\frac{(-1)^n\cos((2n+1)\pi y/(2b))}{(2n+1)^3}$}
For $b>0$ and $-b\leq y\leq b$, the even <quadratic polynomial> $b^2-y^2$ has the displayed expansion in even <Dirichlet eigenfunctions>. Put $k_n=(2n+1)\pi/(2b)$. Their squared <norm> on $[-b,b]$ is $b$. Two <integrations by parts>, using $\cos(k_nb)=0$, give
$$
\int_{-b}^{b}(b^2-y^2)\cos(k_ny)\,dy
=\frac{2}{k_n}\int_{-b}^{b}y\sin(k_ny)\,dy
=\frac{4\sin(k_nb)}{k_n^3}.
$$
Division by $b$ gives the displayed <Fourier coefficients>. The cubic decay gives <uniform convergence>. The half-integer frequencies arise from requiring both evenness and zero values at the endpoints, rather than from a periodic boundary condition. This expansion is useful in <rectangular duct flow>.
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