Quadratic arch expansion in half-integer cosine modes ($b^2-y^2=\frac{32b^2}{{\pi }^3}{\sum }_{n\ge 0}\frac{(-1{)}^n\cos ((2n+1)\pi y/(2b))}{(2n+1{)}^3}$)

For $b>0$ and $-b\le y\le b$, 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 $\cos (k_{n}b)=0$, giveDivision 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.