Fourier transform of the hyperbolic secant (${\int }_{\mathbb{R}}{e}^{-ikx}\mathrm{sech}xdx=\pi \mathrm{sech}(\pi k/2)$)

For real $k$, integrate $e^{-ikz}/\cosh z$ around a rectangle of height $\pi$. Its top-edge value is $-e^{\pi k}$ times the bottom-edge value, and the only enclosed simple pole is $i\pi /2$, with residue $e^{\pi k/2}/i$. The vertical-edge integrals vanish as their real parts tend to positive and negative infinity. The residue theorem gives $(1+e^{\pi k})\tilde{f}(k)=2\pi e^{\pi k/2}$, proving the stated Fourier transform identity. This is an example of a function reproducing its shape under a Fourier transform, up to rescaling.