Fourier transform of a truncated cosine ($\hat{f}(k)=\sin [(k-1)a]/(k-1)+\sin [(k+1)a]/(k+1)$)

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 $k=\pm 1$ are removable, with value $a+\sin (2a)/2$. If $\cos a\ne 0$, 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.