Pictures of some generalized analytical continuations

[Caleb]

My point is just that the functions we are dealing with here only have a measure 0 set of bad points-- the rest of the points are well defined. So, I'm interested if your extension will recover any familar theorems when we weaken the definition of holomorphic in the way I have suggested.

Also, I don't mean to disagree with the work you have done. I'm not imply anything you have is wrong. I'm just suggesting that we look at some details of coherence on the unit circle to see if a given extension is 'good'. I.e., we might be able to classify 'bad' extensions by their bad behaviour on the unit circle. My cursory look at generalized analytical continuation suggests they do something like this-- they look for matching boundary values as one condition for continuation. I think complex differentiablity might be another condition to look for, to shift out bad extensions from the good ones

This is my point also, Caleb. Because we described the singularities at \(|z| = 1\) as the same thing; we have the same result for non-singular points on \(|z| = 1\). My equation will describe the point wise differential properties; but the only data it needs to produce this, is singular data for \(|z| =1\).

I am loving everything you are saying. I'm not going to prove this result for you. But all the things you are worried about; it happens. Every thing works exactly as your imagining. The math is turning out to be incredibly beautiful. But it's very technical. And I want to get everything right. So I'm focusing on writing at 25-30 page paper, which covers all bases.

But, what you see as differential points; can be covered by only looking at the singular points. That's kind of the thesis of what I am saying!

I'm super excited to go deeper into this. But, I should probably shut up. I should just finish the 25-30 pages of work that goes into it. This is a paper; not a post on the tetration forum... I should treat this knowledge with the respect it deserves.

Nonetheless! Keep posting, Caleb! I'm always weirdly angry and upset about how your right, but it doesn't make sense, lmfao.

Sincere Regards, James

Yes sorry-- I will shut up for now to let you work on the paper. I'm super excited to read it, so I'll bide my time doing something else for a few days. Good luck on the writing-- I can't wait to read it!

Thanks again for your work,
Caleb