Fractional Bandwidth of a Gaussian Amplitude Modulated Signal a gaussian modulated sinusoidal signal may be expressed as
$$x(t)=A\cdot e^{(j2\pi ft)}\cdot e^{\left[-\frac{1}{2\sigma^{2}}\cdot(t-t_{0})^{2}\right]}$$
Let's consider the case in which the gaussian peak is at 0 and the total amplitude is 1:
$$x(t)=e^{(j2\pi ft)}\cdot e^{\left[-\frac{1}{2\sigma^{2}}\cdot t^{2}\right]}$$

What is the fractional bandwidth of this function?
I'm asking this question since I think it's a quite used parameter. In fact, there are a proper function of MATLAB and a proper function of Python) which can generate this function, and both ask me to specify:

the sine wave frequency (which is $f$);

the signal fractional bandwidth


I think the last one is necessarily linked to the standard deviation $\sigma$. But by which mathematical relationship? It's a bandwidth in time domain (so the FWHM of the gaussian pulse: $2.35\cdot \sigma$) or the bandwidth in frequency domain (so ???)?
The fact that it's modulated with a sinusoid doesn't change the FWHM bandwidth of your pulse – the $e^{jx}$ function has $\left\lvert e^{jx}\right\rvert\equiv 1$ at every point. That doesn't change the amplitude, so the FWHM of a sinusoid-modulated gaussian is just the same as of the unmodulated gaussian.