Pictures of the Chi-Star

[sheldonison]

Define the sequence \( \zeta_n = \Psi^{-1}(e^{-n} \Psi(x_0)) \) where \( x_0 \) is fixed and is arbitrary, so long as \( \log^{\circ n}(x_0) \to L \). We can write an entire super function of exponentiation as

\( F(z)=\frac{1}{\Gamma(Lz)} (\sum_{n=0}^\infty \zeta_n \frac{(-1)^n}{n!(n+Lz)} + \int_1^\infty (\sum_{n=0}^\infty \zeta_n \frac{(-t)^n}{n!})t^{Lz-1}\,dt) \)
...
\( e^{F(z)} = F(z+1) \), but \( F(z) \neq 0 \), and is not real valued on the real line...  I'm still a little lost about how Kneser does it, but even how it looks, it doesn't seem like it'l apply in this scenario.
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this super function looks locally like \( F(z) = \Psi^{-1}(e^{Lz}\Psi(x_0)) \) ...

Then one can also take the log \( L=\lambda \) of the \( \Psi \) to generate the \( \alpha \) Abel function
\( \alpha(z)=\frac{\ln(\Psi(z))}{\lambda}\;\;\;\alpha(e^z)=\alpha(z)+1 \)
And your \( F(z) \) function is the complex valued superfunction; \( \alpha^{-1}(z)=\Psi^{-1}(e^{\lambda z})\; \)

Then the \( \alpha(z) \) of the real number line is the un-spiraled \( \chi \) function and one could superimpose it on the complex valued superfunction.   These two pictures are exactly analogous to the earlier pictures in the post.  Below is the complex valued superfunction, which is \( \Psi^{-1}(e^{\lambda z}) \) from -3 to +6 on the real axis and -3 to +2 on the Imaginary axis.  The green area was mapped from z near 0 from the \( \Psi^{-1}(z)\approx\ L \) region.  The yellow highlight is real axis from approximately \( \text{Tet}(-2)\to \text{Tet}(6) \) with a gap of about 10^-78 near the singularity at zero and a corresponding gap at other integer values.      

And here is the key showing what real numbers the yellow highlighter refers to.  
   

So now we want to map the yellow region (or colored sections in the 2nd picture) to the Tetration real axis between -2 and +6.  The yellow region can be extended infinitely in both directions.  And then we want to map everything "above" the yellow region to the upper half of the complex plane, while keeping the definition of \( \text{Tet}(z+1)=\exp(\text{Tet}(z)) \)  And that's what Kneser's construction does.  There are a lot more details, like how does Kneser generate such a mapping???   But your \( F(z)=\alpha^{-1}(z) \) function is the intermediate step on the way to real valued Tetration.  

In Jay's-description  and in Sheldon's-2011-post, Jay and I both added a lot of details about the (repeating) singularity.  But focusing on the complicated singularity comes at the expense of making the crucial repeating pattern harder for the reader to see.  You might also want to see  Henryk's-description