The following is a function with $\alpha$ being a real constant
$$f(t) = \frac{\sin(\alpha t)}{\alpha t}.$$
Determine the analytic signal $f_a (t),$ Hilbert transform $\hat{f}(t),$ and the envelope of $f(t)$
Attempt
So, $f(t)$ looks like a $sinc$ function which may be written as
$$f(t)=sinc(\alpha t).$$
The analytic signal is given by
$$f_a (t) = 2 \int^\infty_0 F(\nu) \exp(j 2 \pi \nu t)=\frac{1}{|\alpha|} rect \left( \frac{\nu}{\alpha} \right) \exp(j 2\pi \nu t)$$
Is this right? How can I work out the imaginary part (Hilbert transform) from this? For a sinc it must be something like $1-\cos(t)/t$.