Let $G$ be the Gaussian $$G(t,w,c,h)=w\cdot e^{-\dfrac{(t-c)^2}{2h^2}}$$ for some real parameter $t$ and the real constants $w$, $c$, and $h$. Now, let $F$ be a function defined in terms of $G$, $$F(t,w,c,h,o)=G(t,w,c-o,h)-G(t,w,c+o,h)$$ where $o$ is an additional real constant. We then have that the derivative of $F$ with respect to $t$ is $$F'(t)=\frac{w(t-c-o)\cdot e^{-\dfrac{(t-c-o)^2}{2h^2}} - w(t-c+o)\cdot e^{-\dfrac{(t-c+o)^2}{2h^2}}}{h^2}$$ I want to find the real roots of $F'(t)$. That is, I want to solve $F'(t)=0$ for $t$ where $t$ is real.
I've tried isolating $t$ by hand to no avail. I get stuck already at the initial step when I try to take the natural logarithm of the entire expression in an attempt to dissolve the exponential terms.
The function $F$ models the shape of a disturbance to an electrical signal in a small system that I'm working on. The roots of the derivatives are useful in my later analysis. Specifically, I can use the roots to pair the electrical disturbance with the type of event that provoked it.
Thus far I have been using numerical root finding but I thought that an analytical solution must be available. I've also tried to expand $F$ using both Taylor and Poisson series. Unfortunately, the series expansions quickly diverge from the actual signal for a reasonable number of terms.
You can't factor it because it is a trascendental equation. But you can simplify a bit, and then use numerical methods. You can rewrite the equation as
$$ (t-c-o)e^{-\dfrac{[(t-c)^2+o^2-2o(t-c)]}{2h^2}}-(t-c+o)e^{-\dfrac{[(t-c)^2+o^2+2o(t-c)]}{2h^2}}=0 $$
Then you can factor and cancel $e^{-\frac{(t-c)^2+o^2}{2h^2}}$ to get
$$ (t-c-o)e^{\dfrac{2o(t-c)}{2h^2}}-(t-c+o)e^{-\dfrac{2o(t-c)}{2h^2}}=0 $$
Using hyperbolic sine/cosine, this can be written as
$$ 2(t-c)\cdot\sinh\left(\dfrac{o(t-c)}{h^2}\right)-2o\cdot \cosh\left(\dfrac{o(t-c)}{h^2}\right)=0. $$
If you want, you can make it a little more compact by passing to the hyperbolic tangent:
$$ (t-c)\cdot\tanh\left(\dfrac{2o(t-c)}{2h^2}\right)=o. $$
Once you get here, there's not much algebra can help you with. You are basically trying to solve $\tanh(x) = a/x$. The solutions to this equation (there are two) can only be computed approximately with numerical methods.