(01/19/2021, 03:01 AM)JmsNxn Wrote: I totally get how analysis is the art of approximating, and complex analysis is the art of approximating uniformly. Don't go thinking I'm staring at excel sheets all day though, lmao! To me it's all just the manipulation of symbols and getting a feel for them. I never numerically evaluate; which is probably a bad thing, but as long as the is there, who cares what the graph looks like or what the numbers are. Numbers can lie, can't. If all us analysts were is big calculators approximating, the riemann-hypothesis wouldn't even be a problem. (What are we at now, 200 trillion zeroes found on the critical line? surely! that's enough.) lol.

Hahaha I surely do not depict you and the other analysts of this forum like that. Writing down my questions on your superfunction trick I had to make a disclaimer on notation and I was tempted to include in it a meme about it. At the end in order to be sober I decided to avoid it, but after this provocation I guess I ll post it here.

Back to serious business...

I think I'm close to understanding the general mechanism behind this superfunction tricks that we can see from Kneser's to Tommy's 2sinh, and that you wanted to upgrade to iterated compositions.

Since I'm very ignorant of real and complex analysis here to really appreciate all the difficulties I went more algebraic on it: it is possible that I'm still blind to some fundamental gear of this mechanism.

I give you here a pdf with a LaTex version of the argument and of the "existential questions" that follow

.

I hope you can skim thru it.

2021_01_16_Generalized_superfunction_trick.pdf (Size: 307.64 KB / Downloads: 20)

To give you the summary: I'm breaking the usual method (Kneser's, principal Abel/Schroeder) in two parts: the construction of the sequence and effectiveness of the limit of the sequence to our purposes. I gave indeed a real proof of the first part (in the appendix) but I propose a formalization schema for the second part.

Anyways I'll make a separate post on it on the forum so you and the others can eventually make some constructive criticisms of it.

ADDENDUM: This is just a draft complied very quickly during the last 3 nights... hahah so forgive me for some grammar errors. I'm already seeing 4 typos...