= Fourier transform of a truncated cosine
{c}
{title2=$\widehat f(k)=\sin[(k-1)a]/(k-1)+\sin[(k+1)a]/(k+1)$}
For $f(x)=\cos x$ on $[-a,a]$ and zero outside, the angular-frequency <Fourier transform> follows by writing the cosine as two <complex exponentials>. Combining fractions gives $[2k\cos a\sin(ak)-2\sin a\cos(ak)]/(k^2-1)$. The apparent singularities at $k=\pm1$ are removable, with value $a+\sin(2a)/2$. If $\cos a\ne0$, jumps at the endpoints give an oscillatory $O(|k|^{-1})$ envelope. If $\cos a=0$, the function is continuous at the endpoints but its first <derivative> jumps, giving an $O(|k|^{-2})$ envelope. This is an explicit instance of <Fourier decay from a derivative jump>.
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