& = & \frac{\alpha}{2}(e_{2n}-e_{2n-1})^2. \nonumber \end{aligned}$$

Now comes the crucial fact: when $\alpha \leq e$, the sequence $e_n(\alpha)$ converges to $W(\alpha)/\alpha$, where $W(\alpha)$ is the Lambert function, defined for $\alpha \geq 0$ as the unique real solution of the equation $W e^W = \alpha$. The function $W(\alpha)$ is analytic for $\alpha \geq 0$. When $\alpha > e$, $W(\alpha)/\alpha$ remains a fixed point of the iteration Eq. (\[eq:en\]) but it is unstable: the sequence $\{e_n\}$ is oscillating. However the even subsequence $e_{2n}$ and odd subsequence $e_{2n+1}$ are still convergent. The even limit is strictly larger than the odd limit. We define the functions $A(\alpha)$ and $B(\alpha)$ for $\alpha \geq 0$ by $$\lim_{n \rightarrow \infty} \, e_{2n} =  \frac{B}{\alpha} \quad
  \mathrm{and} \quad \lim_{n \rightarrow \infty} \, e_{2n+1} = 
  \frac{A}{\alpha}.$$ Then $(A,B)$ solve the system $$Ae^B=\alpha, \quad  Be^A=\alpha. 
  \label{eq:ab}$$ For $\alpha \leq e$, the unique solution is $A = B = W$. For $\alpha > e$, the previous solution becomes unstable and $(A,B)$ is the solution of Eq. (\[eq:ab\]) selected by the rule $A < W < B$. This is summarized on Fig. \[fig:abw\].

![ *The special functions $W(\alpha)$, $A(\alpha)$ and $B(\alpha)$, which coincide for $\alpha \leq e$.* []{data-label="fig:abw"}](abw.eps){width="\figwidth" height="0.9\textheight"}

Taking the limit 