Isolate Sigma and FWHM from gaussian

Question

I have the Gaussian:

$a e^{-b^2 (x-c)^2}$

And need to isolate the Sigma and FWHM from it. I believe that

$b = \frac{1}{\sigma^2}$

and

$FWHM = 2.354(\sigma/2)$

However, I need to program this into a system, and of course isolating for sigma from B doesnt produce a single answer, instead it gives me two. And then you end up with two FWHM.

Can anyone tell me what I'm doing wrong?

Answer 1

Note that the standard form of the Gaussian is $$pe^{-\dfrac{(x-q)^2}{2r^2}}$$

In your equation, we have $-\frac{1}{2r^2}=-b^2$, thus $r^2=\frac{1}{2b^2}$, thus $r=\sigma=\pm\sqrt{\frac{1}{2b^2}}$. However, the standard deviation is always positive.

---

Therefore the results are:

$$\sigma=\sqrt{\frac{1}{2b^2}}=\frac{\sqrt{2}}{2b}$$

$$\mathrm{FWHM} = 2 \sqrt{2 \ln 2} \sigma \approx 2.35482\sigma$$