I was studying the underdamped harmonic motion and got curious about the fact that the decreasing exponentials $\pm Ae^{-\gamma t}$ are good approximations only for light damping $(\gamma<<\omega) $. So I've searched and found that $\pm Ae^{-\gamma t}$ are the envelope of the motion (i.e. tangent to points of the curve that represents it) and the points of tangency are not coincident to the maxima and minima of the curve, as shown in the image below (Source: leancrew.com):
The question that came immediately in my mind was "How to find the exponentials that contains all maximum/minimum points?", so I've searched more and found a footnote in Morin's Introduction to Classical Mechanics:
To be precise, the amplitude doesn't decrease exactly like $Ce^{-\gamma t}$, as Eq. (4.16) suggests, because $Ce^{-\gamma t}$ describes the envelope of the motion, and not the curve that passes through the extremes of the motion. You can show that the amplitude in fact decreases like $$Ce^{-\gamma t} \cdot cos\left(tan^{-1}(\gamma/ \tilde{\omega})\right) .$$ This is the expression for the curve that passes through the extremes.
Eq. (4.16): $$\space x(t)=e^{-\gamma t} \left(Ae^{i \tilde{\omega}t}+Be^{-i \tilde{\omega}t}\right) \equiv e^{-\gamma t}C \space cos\left(\tilde{\omega }t+\phi\right)$$ $$\tilde{\omega} \equiv \sqrt{\omega^2-\gamma^2}$$
So, now I know the equation of exponentials that contains all the points of maximum/minimum: $$\pm Ce^{-\gamma t} \cdot cos\left(tan^{-1}(\gamma/ \tilde{\omega})\right) $$
but, after trying a lot, I still couldn't figure out how to deduce them from Eq. (4.16). How can I do this?