How do you take the Fourier transform of $f(x)= \dfrac{1}{\cosh^2(x)}$, using contour integration? (Where the complex Fourier transform of $f(x)$ is defined to be $\hat{f}(k) = \displaystyle \int_{-\infty}^{\infty} f(x)e^{-ikx}\,dx$).
This is for a complex class so I tried expanding the denominator and calculating a residue by using the rectangular contour that goes from $-\infty$ to $\infty$ along the real axis and $i\pi+\infty$ to $i\pi-\infty$ to close the contour (with vertical sides that go to 0). Therefore, I tried to calculate the residue at $\frac{i\pi}{2}$ of $\dfrac{e^{-ikx}}{e^x + e^{-x}}$ which will be give me the answer, but I don't know how to do this.