I was solving an equation in Mathematica and I stepped in the following expression
(2 a A m + h^2 ProductLog[(2 a A E^(-((2 a A m)/h^2)) m)/h^2])/(2 a h^2)
Which is $$\frac{2 A a m+ \hbar^2 W(\frac{2 A a E^{-\frac{2 a A m}{\hbar^2}} m }{\hbar^2})}{2 a \hbar^2} = \frac{A m}{\hbar^2} + \frac{1}{2 a} W\left\{\left(\frac{2 A a E^{-\frac{2 a A m}{\hbar^2}} m }{\hbar^2}\right)\right\}$$
so it's form is $w \,e^{-w}$. I expect, by the little to nothing I know of the Lambert W Function, that $W(w \,e^{+w})=w$ but I do not know what the value of $W(w\,e^{-w})$ would be. Note that if $W(w\,e^{-w})$ is indeed $w$ then the expression simplifies in a very compact way. On the other hand, if it's equivalent to $-w$ it would be reduced to $0$.
Question: what´s the actual value of $W(w e^{-w})$ and how could I have calculated it?
Thanks in advance,
I suppose that plotting $W(x e^{-x})$, you notice that it is a quite strange function. To give you an idea, its expansion around $x=0$ write $$W(x e^{-x})=\sum_{n=1}^\infty (-1)^{n-1} a_n\, x^n$$ the first $a_n$'s making the sequence $$\left\{1,2,4,\frac{28}{3},24,\frac{328}{5},\frac{8416}{45},\frac{173216}{315},\frac {104288}{63},\frac{2878496}{567},\frac{8297792}{525},\frac{517896128}{10395}\right\}$$ which are not in $OEIS$.
If you need to compute its value for a given $x$, you do not have much choice. You will find many subroutines for the computation of $W(t)$.
If you have to solve for $x$ the equation $y=W(x e^{-x})$ (for $y>0$), tak care that there is no solution if $y>W\left(\frac{1}{e}\right)\approx 0.278465$ and otherwise, there are two roots $(0\leq x_1 \leq 1$ and $ 1\leq x_2 <\infty)$.
Edit
If $x>0$, $xe^{-x}$ is small; so, for approximate values, let $t=$xe^{-x}$ and use the series expansion.
You can also have quite good approximations using the $[n+1,n]$ Padé approximants. They will write $$W(t)=t \frac {P_n(t)}{Q_n(t)}$$ I give you below a table of the polynomials $P_n(t)$ and $Q_n(t)$ as well as the maximum error at $t=\frac 1e$
$$\left( \begin{array}{cccc} n & P_n(t) & Q_n(t) & \Delta_n \\ 1 & 1+\frac{t}{2} & 1+\frac{3 t}{2} & 2.20 \times 10^{-3} \\ 2 & 1+\frac{19 t}{10}+\frac{17 t^2}{60} & 1+\frac{29 t}{10}+\frac{101 t^2}{60} & 6.10\times 10^{-5}\\ 3 & 1+\frac{623 t}{190}+\frac{123 t^2}{50}+\frac{1927 t^3}{11400} & 1+\frac{813 t}{190}+\frac{4977 t^2}{950}+\frac{18881 t^3}{11400} & 1.74\times 10^{-6} \end{array} \right)$$
Optimizing the model, a good approximation is $$W(t)=t\frac{1+\frac{5323 }{3340}t+\frac{254 }{1599}t^2}{1+\frac{7205 }{2778}t+\frac{3773 }{3005}t^2}$$ which gives a maximum error significantly smaller than $10^{-7}$.