I know the Hanning window, $w_{Hann}(n) = 1 - \cos \left(2\pi \frac{n}{N} \right)$, is typically applied to discrete Fourier transforms. $N$ being the total number of points in a transient signal that is to be transformed with the DFT. However, I'm trying to have a look at how this window (which is what I use to measure) affects the spectral line shape of the resultant Fourier transform.
For now I simply say my transient signal is a damped oscillator of the form $$f(t) = e^{-t/\tau} \cos(\omega_0 t).$$The Fourier transorm of which is well known to be a Lorentzian profile. However As I measure my data with a Hanning window I want to take the Fourier transform of $$f(t) w_{Hann}(t)$$ This means I need the Hanning window in the time domain, which I assume to be simply $$w_{Hann}(t) = 1 - \cos \left(2\pi \frac{t}{T} \right)$$ where $T$ is the averaging time for my Fourier transform.
Is this a good approach and a sounds assumption I can make with the Hanning window in a time domain rather than for a discrete index of points?