A question concerning uniqueness

[sheldonison]

Oh very very interesting!

So I guess, the definitive property for exponentiation would be:
\( a\cdot a^x =a^{x+1} \) and no other function satisfies this requirement.

I wonder, what exactly differentiates tetration such that there are multiple solutions given for \( ^b a \) when \( a^{^b a} = \,\,^{b+1} a \) is the definitive property...

edit: this reply only applies for bases>eta, because for bases<=eta, the superfunction is real valued, so no Kneser mapping is necessary. For bases<eta, there are multiple superfunctions, one of which is entire, and the solutions are imaginary periodic, so the uniqueness discussion is much more complicated.

The uniqueness problem has basically been solved, and involves the behavior of sexp(z) in the complex plane as imag(z) increases. If you start with the Schroder function for exp(z), then that function has only one solution. Then the inverse Schroder function for exp(z) can be trivially turned into a unique complex valued superfunction for exp(z). Here, I'm using S for the Schroder function.
\( S^{-1}(\lambda z) = \exp(S^{-1}(z)) \),
\( \lambda = L\approx0.318+1.337i \), the fixed point for exp(z).

From there, there is one unique complex superfunction for exp(z). The superfunction has a complex period, and as imag(z) increases, and real(z) decreases, the superfunction decays to the fixed point.
\( \text{superfunction}(z)=S^{-1}(\exp(z L)) \)

Now, from there, there is exactly one unique real valued tetration, which converges to the complex valued superfunction(z+k), where k is a constant, as imag(z) increases. Here, \( \theta(z) \) is a 1-periodic periodic function that decays to a constant as imag(z) increases. \( \theta(z) \) has a singularities for integer values of z.
\( \text{sexp}(z)=\text{superfunction}(z+\theta(z)) \)

Any other different real valued sexp(z) function will have singularities as imag(z) increases, rather than converging to the complex superfunction. \( \text{sexp_{alt}(z)=\text{sexp}(z+\theta_{real}(z)) \), where \( \theta_{real}(z) \) is another different real valued 1-periodic function used to generate the proposed alternate sexp(z) solution from the correct sexp(z) solution. The important thing turns out to be that the alternative sexp(z) function is generated using \( \theta_{real}(z) \) which is required to be real valued. Because it is required to be real valued, it is easy to show that it grows exponentially as imag(z) increases.

There's a little more to it. I wrote a program that calculates theta(z), but it is tricky to rigorously prove the algorithm converges, although I have some good heuristic arguments. Kneser's approach is to use a unique Riemann mapping, which can be shown to be mathematically equivalent to the theta(z) approach, see this post. Kneser's Riemann mapping rigorously guarantees the existence of one unique solution real valued sexp(z). The net result is that there is one unique real valued sexp(z) function, that is well behaved in the complex plane and doesn't have any singularities for real(z)>-2.
- Sheldon