some questions about sexp

[bo198214]

Though tempting for uniqueness this seems not to be the case, the half iterate of exp(x)-1 (which is conjugate to \( e^{x/e} \)) has the following powerseries coefficients at 0:

Code:

0, 1, 1/4, 1/48, 0, 1/3840, -7/92160, 1/645120,...

That Taylor series does not converge (I've seen this one before, it's from the parabolic regular iteration, right?), though. So \( 0 \) is actually a singularity of some kind of \( \mathrm{dxp}^{1/2}_{e^{1/e}}(x) \) (\( \mathrm{dxp}_b(x) = \exp_b(x) - 1 \)) and so there is no Taylor expansion there.

Yayaya, but these are called "asymptotic Taylor expansions". The Taylor expansions of regular \( \mathrm{dxp}^{1/2}_{e^{1/e}} \) at \( x_0\neq 0 \) converge to the given coefficients for \( x_0\to 0 \) though it is not analytic at 0.

But this is sufficient for our case, as for \( x_0 \) close enough to 0 this one derivative must turn negative.