Danger in performing phase shift with Python for a time series

Mathmatical expression

For scientific purpose, we sometime need to shift phase of time series with specfic angles. To acheive this, we can transform the time series into freqeuency domain with $$F(\omega) = \int_{-\omega}^{\omega} f(t) e^{-i \omega t} dt$$

where $F(\omega)$ as Fourier spectrum of signal $f(t)$. Thus, the phase shifted Fourier spectrum $F_{sh}$ is

$$F_{sh}(\omega) = F(\omega) * e^{-i \alpha}$$