Solving tetration for base 0 < b < e^-e

[Gottfried]

So then since neither of those worked, it seems all we're left with is the Ansus formula and the Cauchy integral (but determining the correct contours and asymptotic behavior, now that's the rub...).

I'm also curious: What about \( b = e^{-e} \) exactly? You said it converges slowly, but how do you iterate it at all? What is the asymptotic as the tower \( x \rightarrow \infty \)?

I used a complex fixpoint and could generate the matrices for regular iteration (in context of diagonalization). The fixpoint I used is
t0 = -0.1957457524880764 - 1.691199920910569*I
one of its logarithms is
u0 = 0.5320921219863799 + 4.597158013302573*I
where u0 = log(t0) + 2*Pi*I // log giving the principal branch
With this I create the triangular Bell-matrix and diagonalize.

The series has complex terms and is very difficult to evaluate - I accelerate slow converging series usually with Euler-summation, but the series has complex terms and it seems I need also complex order for Euler-summation. With 128 terms I could at least get results which reproduced the integer iteration to such an approximate that I'm confident that the series can be used in principle.
However, the fractional iterates behave even worse, and two half-iterates reproduce the integer iterate just to two decimals...

The schröder-term s for schr(x') and x'=x/t0 - 1 at x=1 is, according to the last three partial sums of the series (128 terms):

Code:

[126]  -0.4119542792176348+1.439754774257274*I
[127]  -0.4119542792176264+1.439754774257268*I
[128]  -0.4119542792176181+1.439754774257268*I
...

where I assume s~ -0.411954279217... +1.439754774257...*I as correct decimals.
From here we can compute y' = schr°-1(s * u0^h ) and with h=1 I reproduce exp_b°1(1) = b =exp(-exp(1)) to 15 digits exact.
The last three partial sums of the series:

Code:

´
[126]   -1.004456441437337+0.03850266639539727*I
[127]   -1.004456441437337+0.03850266639539727*I
[128]   -1.004456441437337+0.03850266639539727*I
...

From here y' = -1.004456441437337+0.03850266639539727*I, y=(y'+1)*t0 , y-b = -7.405812 E-17 + 2.57894 E-15*I is an acceptable approximation. However - it is only the integer-iteration. I've not yet found en Euler-order or another convergence acceleration which stabilizes the partial sums for the half-iterate to an acceptable degree.

Gottfried