# Function Similar to Lambert $W$ Function

It is known how to solve equations of the form $$e^{-cx}=a_0(x-r)$$ in terms of the Lambert $$W$$-function. Namely, according to Wikipedia, the solution is given by $$x=r+\frac{1}{c}W\bigg(\frac{ce^{-cr}}{a_0}\bigg).$$

Is there a similar way to solve equations of the form $$e^{-cx^2}=a_0(x-r),$$ where say $$a_0,c,r>0$$? In this case, there is indeed a unique solution with positive $$x$$ value, for the left hand side is positive and the right hand side is linear and increasing in $$x$$ and is negative at $$x=0$$. Without the $$r$$, one could reduce to the above solution by squaring. I'd be interested in if there are either special functions that are reasonably well-understood to solve this, or if there are any reasonable approximations on the value of $$x$$ in terms of $$c,a_0,$$ and $$r$$. Thanks!

• Well, the $r=0$ case reduces to the Lambert function if we square both sides. The general case is similar to generalized Lambert functions that invert $R(x)e^x$ for a rational $R$, except here it is only rational in $\sqrt{x}$. Did this come up in an application? – Conifold Jun 27 at 5:07
• No, this is not solvable. – Simply Beautiful Art Aug 9 at 13:56