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On to C^\infty--and attempts at C^\infty hyper-operations
As Sheldon has thoroughly convinced me of non-holomorphy of my tetration. I thought I'd provide the proof I have that it is on the line . I sat on this proof and didn't develop it much because I was too fixated on the holomorphy part. But, I thought it'd be nice to have a proof of .

Now, the idea is to apply Banach's fixed point theorem, but it's a bit more symbol heavy now. We will go by induction on the degree of the derivative. So let's assume that,


And is the sup-norm across some interval . As a forewarning, this is going to be very messy...

Now to begin we can bound,

And that next,


So, we ask you to put on your thinking cap, and excuse me if I write,

And by the induction hypothesis,

Which is because these terms are made up of finite sums and products of and these are said to be summable.  Now the proof is a walk in the park.

Where, we've continued the iteration and set and for , and for (but we're tossing this away because we know it's differentiable). Therefore,

Of which, I've played a little fast and loose, but filling in the blanks would just require too much tex code.

EDIT: I'll do it properly as I correct my paper and lower my expectations of the result.


As to the second part of this post--now that we have out of the way, we ask if we can continue this iteration and get pentation.  Now, will certainly be and so it's a well defined bijection of . So, first up to bat is to get another phi function,

This will be (it'll be a bit trickier to prove because we aren't using analytic functions, but just bear with me). And it satisfies the equation,

By now, I think you might know where i'm going with this.

And now I'm going to focus on showing this converges... Wish me luck; after being trampled by this holomorphy I thought I'd stick to where things are nice--no nasty dips to zero and the like...

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On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/08/2021, 12:12 AM

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