My equation is:
$$ \frac{w}{2\sqrt{\Pi \sigma^2}} e^{\frac{-(x-\delta)^2}{4 \sigma^2}}=x $$
I am struggling to solve it fo $x$ knowing that: $ \sigma, w,\delta$>0.
Probably, it could be solved somehow using the Lamber-W function.
Help would be appreciated.
Maybe this helps someone, but there is a solution for $\delta = 0$. Firstly, let $\alpha = \frac{w}{2\sqrt{\Pi \sigma^2}}$. Then
$\alpha \exp(-\frac{x^2}{4\sigma^2}) = x$
$\alpha = x \exp(\frac{x^2}{4\sigma^2})$
$\alpha^2 = x^2 \exp(\frac{x^2}{2\sigma^2})$
$\frac{\alpha^2}{2\sigma^2} = \frac{x^2}{2\sigma^2} \exp(\frac{x^2}{2\sigma^2})$
$x^2 = 2 \sigma^2 W(\frac{\alpha^2}{2\sigma^2})$
$x = \sqrt{2 \sigma^2 W(\frac{\alpha^2}{2\sigma^2})}$
Note that negative answer to the square root is spurious, it appears because we squared the expression earlier.
Unfortunately, this trick does not work out of the box when $\delta>0$. Maybe there is another clever trick, maybe it is unsolvable, I don't know