TITLE: Inverse Function of $(1-e^{-t})t$
QUESTION [6 upvotes]: I'm interested if there is some hope to obtain the inverse of
$$ f(t) = (1-e^{-t})t $$
for $t$ positive. If there is a formula I suspect that the Lambert W function will be involved on it. Clever approximations are also welcomed (not looking for local ones though ;).

REPLY [3 votes]: For large $t$ $f(t)\sim t$, while for small t $f(t)\sim t^2$. Then for large $y$ $f^{-1}(y)\sim y$ and for small $y$ $f^{-1}(y)\sim \sqrt y$. Using this as a first approximation in Newton's method gives the following approximation:
$$
f^{-1}(y)\sim h(y)=\begin{cases}
\sqrt{y}-\frac{-y-e^{-\sqrt{y}} \sqrt{y}+\sqrt{y}}{e^{-\sqrt{y}}
   \sqrt{y}-e^{-\sqrt{y}}+1}& 0<y<1,\\
y+\frac{e^{-y} y}{e^{-y} y-e^{-y}+1}& y\ge1.
\end{cases}
$$
This is the graph of $f^{-1}(y)-h(y)$: