For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Does the sequence of functions $(w_n)$ converge uniformly to $W$ on $[0, \mathrm{e}]$?
For every positive real $x$, we consider the function $\phi_x$ defined by
$$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$
and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by
$$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$
Let $W$ be the Lambert function defined in Part I. Does the sequence of functions $(w_n)$ converge uniformly to $W$ on $[0, \mathrm{e}]$?