I am trying to figure out the width-at-half-maximum for a speical case of the difference of Guaussians. In this scenario, I start with a standard formula for a Gaussian:
where x0, is the location of the peak, S is the spread, and Gmax is the amplitude. However, then I take a difference of two, assuming that x0 is 0 for both, that Gmax is unity for both, and the only thing free to vary between them is the spread.
Here I also assume that S_a > S_d, and that all values (including x) are real positive numbers. In this case, I (believe) I always get a peak function that rises from y = 0 at x = 0 to some peak, and then falls back to y = 0 at infinity.
Differentiating fDOG and solving for y = 0, I can find the time of the peak of this function:
I can then find the amplitude at the peak by substituting tpeak for x in fDOG. However, what I would like to do now is is find the (two) points on x where y = fDOG(tpeak, S_a, S_d)/2. Is this possible?