I have to calculate Fourier serie and Fourier transform of periodic extension, of period $T=2$ of the following function:
$ x_0(t) = e^{−2|t−1|}(u(t) − u(t − 2))$
I derived two times and then transformed. The result is the following:
$X_0(\omega)=\frac{\frac{2}{e^2}+\frac{2}{e^2}e^{-2j\omega}-\frac{1}{e^2}j{\omega}e^{-2j\omega}+\frac{1}{e^2}j\omega-4e^{-j\omega}}{-4-\omega^2}$
Now I know that I have to sample in $(\omega_ok)$ with $\omega_0=\pi$
$X_0(k\pi)=\frac{\frac{2}{e^2}+\frac{2}{e^2}e^{-2jk\pi}-\frac{1}{e^2}j{k\pi}e^{-2jk\pi}+\frac{1}{e^2}jk\pi-4e^{-jk\pi}}{-4-(k\pi)^2}=\frac{4(-1)^k-\frac{4}{e^2}}{4+(k\pi)^2}$
discontinuity in $\pm(\frac{2j}{\pi})$
I would like to know how to proceed to find coefficients, Fourier serie and Fourier transform with these complex discontinuities