I assure you it's not a mistake. But I did misspeak a bit. Observe,
\(
f(x) = e^{\displaystyle x-1+e^{\displaystyle x-2 +e^{...x-n+f(x-n)}}}\\
\)
Now \( f(x-n) \to 0 \) as \( n\to\infty \), and therefore,
\(
f(x) = e^{\displaystyle s-1+e^{\displaystyle s-2 +e^{...}}} = \phi(x)\\
\)
We can actually decrease this condition to \( f(x-n) \to C \); it doesn't matter because the limit always converges to \( \phi \). I like to then categorize the uniqueness as being asymptotic to \( e^{s-1} \) which is equivalent to \( f(x) \to 0 \) but it sounds a bit nicer. This is the beauty of always using Banach in everything I do, lol.
As to the originality of this thesis, I assure you it's 100% mine. It falls under the category of infinite compositions, and not to speak too harshly, the field is dead. No one does anything. I keep a correspondence with the only other person who seems to be actively working on it, Dr. John Gill. And he derived similar results to mine; but nothing of holomorphy; and nothing on differentiability. Actually, he did very similar things to what I'm doing--but with continuous functions mostly. He also says nothing about the functional equation aspect. And the summability criterion are all my own; though John uses similar constructs (I tried to boil it down into a single condition, the convergence of a sum); he sort of talks about different types of conditions. My condition is definitely weaker though and encompasses most of John's. Plus holomorphy is a cheap write off, lol.
To find my papers just google James David Nixon arXiv, I just publish them on arxiv as I'm too lazy to be bothered with usual journals. Always such a headache and I doubt I'd gain much prestige from the journals I could get in anyway, lol. Plus they always ask for a publishing fee of like 200$ and I ain't got spare cash like that, lol. This is the one that started it all though,
[/url][url=https://arxiv.org/abs/1910.05111]\( \Delta y = e^{sy} \), Or How I Learned To Stop Worrying and Love The Gamma Function
It's written loosely, I like to play with language <_< when I probably shouldn't. But I believe expository style math papers are easier to digest, especially when introducing notation and the sorts.
And on what this tetration's called; I hardly care about what it's called; call it what you want. I just call it a tetration function. I need a good uniqueness condition first.
I'm suspicious that this solution is actually Kneser's solution; hard to explain why, but I'll have to do more manipulations to argue that.
Thanks, Tommy. I do believe I have come a long way, and come into my own; a lot of my old posts make me cringe a fair amount. Luckily my internet footprint was fairly small and I didn't say something too stupid to too large an audience. I eventually learned to just shut up and pick up a book, lol
Regards, James
\(
f(x) = e^{\displaystyle x-1+e^{\displaystyle x-2 +e^{...x-n+f(x-n)}}}\\
\)
Now \( f(x-n) \to 0 \) as \( n\to\infty \), and therefore,
\(
f(x) = e^{\displaystyle s-1+e^{\displaystyle s-2 +e^{...}}} = \phi(x)\\
\)
We can actually decrease this condition to \( f(x-n) \to C \); it doesn't matter because the limit always converges to \( \phi \). I like to then categorize the uniqueness as being asymptotic to \( e^{s-1} \) which is equivalent to \( f(x) \to 0 \) but it sounds a bit nicer. This is the beauty of always using Banach in everything I do, lol.
As to the originality of this thesis, I assure you it's 100% mine. It falls under the category of infinite compositions, and not to speak too harshly, the field is dead. No one does anything. I keep a correspondence with the only other person who seems to be actively working on it, Dr. John Gill. And he derived similar results to mine; but nothing of holomorphy; and nothing on differentiability. Actually, he did very similar things to what I'm doing--but with continuous functions mostly. He also says nothing about the functional equation aspect. And the summability criterion are all my own; though John uses similar constructs (I tried to boil it down into a single condition, the convergence of a sum); he sort of talks about different types of conditions. My condition is definitely weaker though and encompasses most of John's. Plus holomorphy is a cheap write off, lol.
To find my papers just google James David Nixon arXiv, I just publish them on arxiv as I'm too lazy to be bothered with usual journals. Always such a headache and I doubt I'd gain much prestige from the journals I could get in anyway, lol. Plus they always ask for a publishing fee of like 200$ and I ain't got spare cash like that, lol. This is the one that started it all though,
[/url][url=https://arxiv.org/abs/1910.05111]\( \Delta y = e^{sy} \), Or How I Learned To Stop Worrying and Love The Gamma Function
It's written loosely, I like to play with language <_< when I probably shouldn't. But I believe expository style math papers are easier to digest, especially when introducing notation and the sorts.
And on what this tetration's called; I hardly care about what it's called; call it what you want. I just call it a tetration function. I need a good uniqueness condition first.
I'm suspicious that this solution is actually Kneser's solution; hard to explain why, but I'll have to do more manipulations to argue that.
Thanks, Tommy. I do believe I have come a long way, and come into my own; a lot of my old posts make me cringe a fair amount. Luckily my internet footprint was fairly small and I didn't say something too stupid to too large an audience. I eventually learned to just shut up and pick up a book, lol
Regards, James
