Funny pictures of sexps

[bo198214]

I'm not particularly familiar with Dmitrii's "Cauchy slog", though looking at the images I'm not really seeing a noticeable difference between that and the intuitive slog.

Thatswhy I am pointing it out. You also mentioned these singularities hidden under the principal branch. Dmitrii describes a completely similar structure for his cslog here:
http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractal.pdf
page 10, figure 4.

As for Kneser's construction, I gave an initial look and was quickly overwhelmed; it will take me some time to properly decipher it, so I can't really comment on it yet.

Did you read the original article? You know I summarized his article here. Kneser exactly showed the construction of a real-valued superlogarithm from the regular one. The regular slog maps the upper halfplane to some infinite region D (because it has singularities on the real axis at \( \exp^{[n]}(0) \)). Now from the Riemann mapping theorem for any two regions (except the whole plane) there is a biholomorphic mapping between them. Kneser uses this biholomorphic mapping \( \phi \) to map D back to the upper halfplane. And shows that \( \phi(z+1)=\phi(z)+1 \), hence also \( \operatorname{kslog}(z)=\phi(\operatorname{rslog}(z)) \) satisfies the Abel equation \( \operatorname{kslog}(\exp(z))=\operatorname{kslog}(z)+1 \).

kslog maps G (the region bounded by L1 and exp(L1)) biholomorphically to an imaginary unbounded region. I showed that this condition (about which we would agree that the islog also satisfies it) is a uniqueness criterion here. So why am I arguing against its uniqueness? Because I finally accept statements only with proof!