Migration of inflection points in y = b # x, for e^(1/e) < b < +oo

[andydude]

"In the domain b >= e ^ (1/e) [b >= Eta], the plot of the real branch of y = b # x [b-tetra-x] show a smooth one-valued always increasing real function. Its first derivative is always >= 0 and its second derivative is a real function with a zero at x = xf. The value of xf against b is a smooth always decreasing function, with a horizontal asymptote for b -> + oo, at x = -1 and a vertical asymptote for b -> Eta, where x = + oo".

Agreed! You really don't need any proof of the asymptotes (at least for me), since I've worked with tetration enough to visualize it. But the two values I would like to see are \( f(2) \approx 0 \) and \( f(e) \approx 0.5 \), which would be interesting if there were a proof of either of these.

The problem is that the perturbations of the Andydude tests in the upper-left corner of his figure (lower bases) are strange and probably due to computing overflow. ... Either Andrew's derivation procedures were not accurate enough, or they went overflow or the function itself to which they were applied is not sufficiently appropriate, for low bases.

No, it is not due to overflow. As Jay D. Fox noted (I don't remember where), My approximations are making the assumption that \( \left[\frac{d^n}{dx^n} \text{slog}_b(x)\right]_{x=0} \) is zero for all n greater than the approximation number (which now that I think about it, is actually true for n=infinity, although I don't know how to prove this), and so its not so much that my computing is approximate (my computations are exact for the graph I attached above), the problem is that the coefficients are inexact, and this inaccuracy in the coefficients causes inaccuracies in the whole computation. But since the coefficients of my approximations are rational for all base b such that \( b = e^q \) where q is rational, they can be represented exactly (technically as pairs of big-ints) but this rational number coefficient is only an approximation to the actual real number coefficient (I'm guessing they are real), so the error propagates through the exact computations.

Does this make sense to you?

If it does, then I could confuse you even more: There is also a possibility that since I am using so many series, one of them was inverted at a point that was outside its radius of convergence. But I never did that analysis, so I should probably do that before coming to any conclusions.

Andrew Robbins