Base 'Enigma' iterative exponential, tetrational and pentational

[Cherrina_Pixie]

Code:

base          1.63532449671527639934534
fixed point   1.63698995729105702725471 + 1.52462061222409266551551*I

...

pentation base        1.63532449671527639934534
pentation(-0.5)       0.540797083552851488750873
sexp fixed point      -1.64087257571659334856123
sexp slope at fixed   4.80600575430175169638439
pentation period      4.00236960853189042690462*I
pentation singularity -1.64567803532871618956796 + 2.00118480426594521345231*I
pentation precision, via sexp(pent(-0.5))-pent(0.5)
                      -8.68755408487736716409823 E-22
complex sexp Taylor series centered at 3.0885322718067176544821807826411

sexp base, sexp(upfixed)=upfixed 1.6353244967152763993453446183062
init;loop iterations required    7
upfixed, parabolic fixed point   3.0885322718067176544821807826411
sexp'(upfixed) base B            1.0000000000000000000000000000000
sexp(upfixed)-upfixed error      9.7027744501722372002703206199650 E-33

? sexp(3.000)
%210 = 3.0022195105134555312059775955040
? pent(2.000)
%184 = 2.0088543076992014631570864956219

Pentation code from http://math.eretrandre.org/tetrationforu...34#pid5334

Let \( \nu \) 'enigma' (Greek αίνιγμα, aínigma) be the base at which the tetrational has a parabolic fixed point, as shown above. This is analogous to \( \eta = e^{1/e} \), the base at which the exponential has a parabolic fixed point.

The interesting things about this base are that several values shown above are close to \( \nu \), often to within one part in one hundred or less, and that some others come to near-integers. Ignoring signs, the real part of the fixed point of the exponential, the lower fixed point of the tetrational (sexp) and the real part of the pentation singularity differ from \( \nu \) by at most 0.010456, roughly one part in 95. Also, the pentation period, the upper fixed point of the tetrational and the fixed point of the pentational aren't so far off from 4.00, 3.00 and 2.00, respectively.

Furthermore, base \( e \) tetration and pentation give values close to \( \nu \):

Code:

? exp(1/2)
%42 = 1.6487212707001281468486507878142
? sexp(1/2)
%43 = 1.6463542337511945809719240315921
? pent(1/2)
%44 = 1.6323247404360631184869762532583
? pent(-3)
%45 = -1.6363583542860289796292230421033

Are these all 'coincidencies' or is there some good reason for the minute differences between these values? Any reason for it seems rather obscure, it's like a puzzle, a mystery to figure out. That is why I call that number, the 'enigma constant'.

Perhaps one can use base \( \nu \), instead of base \( \eta \) or \( e \) to evaluate non-integer iterations of exponential. There's no real fixed point of base \( \nu \) exponential so the half-iterate and such should be real-analytic on the entire domain

       
\( f_1(x) = \nu^x, \ f_2(x) = \exp_{\nu}^{\circ 1/2}(x), \ g(x) = \textrm{sexp}_\nu(x) \)

   
\( h(x) = \textrm{pent}_\nu(x) \)