Cauchy window 1/(1+(2 alpha x)^2) on [-1/2 1/2] (default alpha 3; exact rationals)
CauchyWindow[x]
CauchyWindow[x, α]
Plot[CauchyWindow[x], {x, -1, 1}, Exclusions -> None]
→ -Graphics-Plot3D[CauchyWindow[x]CauchyWindow[y], {x, -1, 1}, {y, -1, 1}, PlotRange -> All, Exclusions -> None]
→ -Graphics3D-CauchyWindow[0.1]
→ 0.7352941176470588