Pictures of the Chi-Star

[sheldonison]

That's even more enlightening! That's a great intuitive way of interpreting it. That's the best way I can interpret it so far, I just get the intuition of the matter. I definitely couldn't reproduce the proof, or teach a class on the proof, but I'm really starting to get how it works. It's so inventive it's mindblowing.
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Thanks a lot for these descriptions, Sheldon. That's why I've come here since highschool. Such a nice and "no question is a bad question" forum. I can't believe how much more rigorous and well-founded my arguments about hyper-operators have become since I've come here.  Too bad it's been less active lately.

Plus it's unreal that that tiny region becomes the half plane and it preserves the composition by \( e^z \) and it solves tetration. Who on god's earth could come up with that?

Thanks a lot for your comments James!

My gut says very few folks understand Kneser's real valued Tetration construction.  Edit it turns out I still only partially understand Kneser too!  When I started this post, I wasn't expecting to try to explain Kneser at all.  I just figured I would post some pretty pictures of the Chi-star contour, and the inverse Schröder function.  The Schröder function and its inverse seems like a "safe" mathematically rigorous picture, and then the reader could imagine Kneser somehow finding a way to unspiral the Chi-star to generate real valued Tetration.  Maybe I made Kneser and real valued Tetration base(e) a little bit more accessible. 

Then the obvious next step is to explain \( z+\theta(z) \) where theta(z) is the 1-cyclic function I originally mistakenly thought was Kneser's Riemann mapping.  We start with this definition of the \( \theta(z) \)  from the RiemannMapping unit circle function.  The \( z+\theta(z) \) approach is mathetmatically equivalent to Kneser's approach, but it is different.

\( z+\theta(z)=\frac{\ln(U(e^{2\pi i z}))}{2\pi i } \)

Remember that U(0)=0, and it is analytic with a radius of convergence of 1.
\( U(z)=a_1 z+a_2 z^2 + a_3 z^3 ... = \sum_{n=1}^{\infty}a_n z^n \)

The equation for z+theta starts by taking the logarithm and dividing by 2pi i so lets rearrange terms a little before substituting \( z \mapsto e^{2\pi i z} \)

\( \frac{\ln(U(z))}{2\pi i}=\frac{1}{2\pi i}\ln\Big(z\cdot a_1\cdot(1+\sum_{n=1}^{\infty}\frac{a_{n+1}z^n}{a_1})\Big)=\frac{1}{2\pi i}\cdot\Big(\ln(z)+\ln(a_1)+\ln\Big(1+\sum_{n=1}^{\infty}\frac{a_{n+1}z^n}{a_1}\Big)\Big) \)


Since ln(1+x) has a simple formal power series, the right hand section also has a formal power series; \( b_1..b_n \) with a radius of convergence of 1.  b0 is the constant term.  I calculated b1 and b2, the first couple of terms of the formal power series.
\( \frac{\ln(U(z))}{2\pi i}=\frac{\ln(z)}{2\pi i}+\sum_{n=0}^{\infty}b_n z^n\;\;\;b_0=\frac{\ln(a_1)}{2\pi i}\;\;\;b_1=\frac{a_2}{2\pi i a_1}\;\;\;b_2=\frac{1}{2\pi i}\Big(\frac{a_3}{a_1}-\frac{a_2^2}{2a_1^2}\Big)... \)

If we substitute \( z\mapsto e^{2\pi i z} \) into this equation for z+theta then the 1-cyclic theta mapping is immediately obvious.
\( \frac{\ln(U(e^{2\pi i z}))}{2\pi i}=z+\theta(z);\;\;\;\theta(z)=\sum_{n=0}^{\infty}b_n e^{2n\pi i z} \)

This is an valid alternative way of viewing Kneser's Tetration construction where theta(z) is a 1-cyclic function which vanishes to a constant as \( \Im(z)\to\infty \), and has a singularity at integer values of z, but is otherwise analytic in the upper half of the complex plane.  So what my kneser.gp pari-gp program does iteratively calculate theta(z), which is computationally much much easier than calculating a Riemann mapping.

\( \text{Tet}(z)=\alpha^{-1}(z+\theta(z))\;\;\; \) Kneser's Tetration in terms of the complex valued inverse Abel superfunction and theta
\( \text{Tet}(z)=\alpha^{-1}(z+\theta(z))=\Psi^{-1}(\lambda^{z+\theta(z)})\;\;\; \) Kneser's Tetration in terms of the inverse Schröder and the 1-cyclic theta mapping.  
And here is one last complex plane graphing image, of Kneser's Tetration base e, from -3 to +12 on the real axis, and +/-3 on the imaginary axis.