Question about Hilbert transform of a cosine signal In the paper entitled "On instantaneous frequency", it is claimed that the Hilbert transform of a signal in the form of $A(t)cos(\phi(t))$ would result in $A(t)sin(\phi(t))$, where $A(t)$ is instantaneous amplitude and $\phi(t)$ is the phase function. 
As an example, consider the following:
$$t = [0, 20],\;\textrm{time interval in seconds, sampled each 0.01 sec}$$
$$A(t) = e^{-0.1(t - 10)^2},\;\textrm{instantaneous  amplitude}$$
$$\omega = [0.1,\ 0.5,\ 1.0,\ 1.5],\;\textrm{angular  frequency  in rad/sec}$$
$$f(t) = A(t)\cos(\omega t)$$
$$g(t) = A(t)\sin(\omega t)$$
$$h(t) = \frac{1}{\pi}P\int_{-\infty}^{+\infty}\frac{f(\tau)}{t - \tau}dt,\;\textrm{Hilbert  transform  of } f(t)\textrm{,  where } P  \textrm{ is the   Cauchy  principal  value}$$
It is expected that $g(t)$ and $h(t)$ be the same according to the fact claimed in mentioned paper. I used function "hilbert" of MATLAB to compute Hilbert transform of $f(t)$ for aforementioned $\omega$ values. Results show that for $\omega$ close to zero, $g(t)$ and $h(t)$ are different and as $\omega$ gets higher, the difference decreases. I was wondering if I am making a mistake or something is wrong with function "hilbert" of MATLAB? 
The following figure shows the results:
Make sure that you understand the conditions under which

$$\mathcal{H}\big\{A(t)\cos(\omega_0 t)\big\}=A(t)\mathcal{H}\big\{\cos(\omega_0 t)\big\}=A(t)\sin(\omega_0t)\tag{1}$$

holds. Eq. $(1)$ holds if $A(t)$ is a low-pass signal with a cut-off frequency smaller than $\omega_0$. This implies that $A(t)\cos(\omega_0t)$ is a band-pass signal with no energy at DC. Eq. $(1)$ is a special case of Bedrosian's theorem.

In your example, $A(t)$ is not band-limited, and, consequently, $(1)$ doesn't hold. However, if you increase the modulation frequency $\omega_0$, the overlap of the spectra of $A(t)$ and of the carrier becomes smaller, because $A(t)$ does have a low pass character. So for large $\omega_0$, Eq. $(1)$ is approximately satisfied, which is exactly what you see in your plots.

Also take a look at this related answer.