Base -1

[sheldonison]

Here is what analytic sexp base(-1) looks like at the real axis.

Wonderful!

Did you tried negative bases close to zero?

I generated a negative tetration base fairly close to zero, with good precision. This is as close as I can get to Kneser analaytic tetration near base(0) so far with my new Abel function program. And its a very artistic tetration base!
Tetration base b~=-0.135335 \( \;\;b=-\exp(-2); \;\;y \mapsto b^y \)
I calculated this using my new pari-gp program (which I will publish soon), which generates the Abel function for iterating \( z \mapsto \exp(z) -1 + k \) for many arbitrary complex values of k. For the tetration base of interest,
\( k = \ln(\ln(b))+1 \; = \; \ln(-2 + \pi i) + 1 \; \approx \; 2.31484985600431 + 2.13770783173591i \)
\( z \mapsto \exp(z) + \ln(-2 + \pi i)\;\; \) which is conjugate or equivalent to the slog/sexp base above. Call the Abel function \( \alpha(z) \). Then if you can take its inverse, \( \alpha^{-1}(z) \) then a reasonably straightfoward, linear equation gives you the desired sexp(z) function.

\( \text{sexp}_b(z) \; = \;
\frac{\alpha^{-1}(z+1+\alpha(k-1))-(k-1)}{\exp(k-1)}\; =\;
\frac{\alpha^{-1}(z+1+\alpha(\ln(-2+\pi i)))-\ln(-2 + \pi i)}{-2 + \pi i}\;\; \)

And here is the resulting sexp function, for b=-exp(-2)~=-0.135335, after taking the inverse of the Abel function. Notice the beautiful 6-cycle attracting fixed point rainbow! The two primary fixed points are both repelling, with the upper fixed point Period1~=2.327+0.182i, and the lower fixed point Period2~=2.357-1.966i The graph goes from -3 to +8 at the real axis, and +2i to -4i at the imaginary axis. I also added a similar complex plane graph to base(-1) to my earlier post; comparing the two plots is interesting.
   

Here is the Abel function, showing the logarithmic branch singularity at the two repelling fixed points,
L1 = 0.209297833082953879535 + 2.68321717005655097610i
and L2 = 1.31091514181888952589 - 1.57185702066772588857i
The Abel function is much more well behaved, especially between the two fixed points where Henryk's uniqueness proof would hold. So then, starting with Tetration base(e), this function is the same function as if you slowly modify the base, going above any singularities, and then get to b~=-0.135335.
I arbitrarily set \( \alpha(\frac{L1+L2}{2})=0\;\; \) and center the Abel function series exactly between the two fixed points. Of course, taking the inverse of the Abel function can be difficult, and it took me several weeks to get it to work. In principle, for sexp(z) you add any integer value n to z, and then find the corresponding \( \alpha^{-1}(z+n+1+\alpha(k-1)) \) in the well behaved region of the Abel function, nearest the line connecting the two fixed points. Then iterate \( \exp(z)-1+k \) or its inverse as needed.
   

And the Taylor series of the sexp function:

Code:

{sexp_bmem2= 1.00000000000000
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+x^ 2* (-2.16809727173656 + 4.19637865176037*I)
+x^ 3* (-7.35090243487976 + 1.04003832671689*I)
+x^ 4* (-7.76932973206147 - 7.27048735575206*I)
+x^ 5* ( 0.589449327914730 - 14.2870020608798*I)
+x^ 6* ( 14.0292409179381 - 11.7028472515246*I)
+x^ 7* ( 22.2709683173421 + 2.66099963739772*I)
+x^ 8* ( 16.2056377985895 + 21.1440445300114*I)
+x^ 9* (-4.00052427101099 + 30.5039203303803*I)
+x^10* (-27.2329501197431 + 21.4708476779723*I)
+x^11* (-38.0970145296299 - 3.63291240324545*I)
+x^12* (-27.3860956401045 - 31.1078116413744*I)
+x^13* ( 1.06541844699315 - 44.1203652438513*I)
+x^14* ( 31.9899439305810 - 33.4505698061264*I)
+x^15* ( 47.7376209547412 - 3.59516873972375*I)
+x^16* ( 38.8744926788335 + 29.6476765419447*I)
+x^17* ( 9.70089125969798 + 48.3842658339320*I)
+x^18* (-24.4212790820708 + 42.7939493910419*I)
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+x^101* ( 7.52305762012355 E-6 - 2.52645938199225 E-5*I)
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+x^103* ( 1.57727846421572 E-5 - 1.10322737968916 E-6*I)
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+x^111* ( 9.08444692415682 E-8 - 1.96010184377976 E-6*I)
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+x^147* ( 3.97059930309326 E-11 + 7.06141823050801 E-11*I)
+x^148* (-4.65389353106001 E-12 + 6.01633108838619 E-11*I)
+x^149* (-2.77642429168665 E-11 + 3.53084921390022 E-11*I)
+x^150* (-3.17621680179065 E-11 + 1.03698476509002 E-11*I)
}

For completeness, here is the graph at the real axis.