A first hires look at tetration \(\lambda = 1\) and \(b = e\)

[sheldonison]

I've been making some different graphs of \(\beta\) and I got a good one to share.

...
All of these will be committed towards the asymptotic thesis of the beta function. Which is that the beta method approaches at least an asymptotic expansion at each point, as opposed to a Taylor series. This is compatible with everything I have been saying, and additionally compatible with Sheldon's work. This paper will entirely focus on ASYMPTOTIC behaviour. Which looks like tetration; but if you try and make it tetration, expect a good amount of errors.

very nice.  I spent a bit more time on b=sqrt(2), and I believe it converges it can be proven to converge analytically so long as the imaginary period of lambda is less than the imaginary period of the attracting fixed point=2.  So lambda=1 would be analytic, but lambda=0.3 would not converge, since
\(\frac{2\pi i}{0.3}>\frac{2\pi i}{\ln(\ln(2))}\)