Why bases 0<a<1 don't get love on the forum?

[sheldonison]

What is the justification to demand that the derivative at zero must be zero?

Every time I tried to develop a Taylor series of tetration, I found that there is ever a degree of liberty, in choosing the derivative at 0.

I could have been more clear; that boundary condition, tet'(-1,0)=0, seemed to work to get a seed from which a Kneser bipolar mapping seems plausible, based on the behavior going smoothly to the pair of conjugate fixed points, in a path to -infinity, from just above and below the real axis. My bonus question, was pretty much the same as your question, "Is it possible that we derive the requirement that tet'(-1)=0 from the requirement that the kneser mapping go to the pair of conjugate fixed points in the complex plane?". edit perhaps tet'(0)=0 is not a requirement for a Kneser solution. Then there may be an infinite number of Kneser solutions with different derivatives... don't know.

edit#2 the solution isn't upside down; For \( b=\exp(-e) \) in the lower half of the complex plane, \( L_{-}\approx-0.195745752488076 + 1.69119992091057i \) and in the upper half of the complex plane \( L_{+}\approx-0.195745752488076 - 1.69119992091057i \) The lower half of the complex plane should be somewhat like the lower half of http://math.eretrandre.org/tetrationforu...0&pid=6748 post#10 and in the upper half, will be its conjugate.

By the way, Andrew's slog won't work, since the Abel function has singularities where sexp'(z)=0. The Kouznetsov's algorithm might work, though from previous experience with b=exp(-e), the function doesn't behave that nicely on the path to the fixed point. The algorithm most likely to converge is the Kneser/theta mapping algorithm. So, now that I have this initial piecemeal seed, the next step would be to write a program that would iterate, generating the Kneser/theta mapping, from the initial seed, and the two complex valued Koenig's solution. This would let us iterate and generate results of arbitrarily high precision. Its not a trivial undertaking, and will require several days of work; probably more like weeks. At the end, running the program will take less than 30 seconds for arbitrary bases between 0 and 1, and then you can instantaneously get very accurate result, and can calculate the function anywhere in the complex plane, so you can make pretty complex plane graphs too. As imag(z) increases, or decreases, the function will go to the complex conjugate pair of fixed points, and will converge to one of the Koenigs solution's in the upper half of the complex plane, and the conjugate Koenig's solution in the lower half of the complex plane ...

Anyway, assume the Kneser/theta mapping converges, then the conjectured uniqueness criteria is that the solution will be analytic in the upper half of the complex plane and the lower half of the complex plane, and at the real axis for z>-2, and converging to the two complex valued Koenig's solutions in the upper/lower halves of the complex plane. The uniqueness criteria and proofs that Henryk has written rely on the slog/Abel function (and the sexp function) both being analytic on a sickle, between the two fixed points. But since the slog/Abel function has singularities where sexp'(z)=0 ... I don't know how one might apply that proof technique to the solution at hand.