Let $x$ be a positive real. I want to find a pair of analytic functions $s(x),d(x)$ such that $s(d(x)) = x$ and
$ s(\exp(d(x)))$ ~ $ x + 2 $
More presicely I Also want :
$$ \lim_{x \to \infty} s(\exp(d(x))) - x - 2 = 0$$
Polynomials seem to fail as do polynomials of exp or ln.
Maybe try Lambert-W ?
Or do I want THE impossible ?