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 On to C^\infty--and attempts at C^\infty hyper-operations JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 02/08/2021, 12:12 AM (This post was last modified: 02/08/2021, 03:55 AM by JmsNxn.) As Sheldon has thoroughly convinced me of non-holomorphy of my tetration. I thought I'd provide the proof I have that it is $C^{\infty}$ on the line $(-2,\infty)$. 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 $C^{\infty}$. 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, $ \tau^{(k)}(t) : (-2,\infty) \to \mathbb{R}\,\,\text{for}\,\,k \sum_{m=1}^\infty ||\tau^{(k)}_{m+1}(t) - \tau^{(k)}_{m}||_{a \le t \le b} <\infty\\$ Where, $ \tau^{(k)}_0(t) = 0\\ \tau^{(k)}_m(t) = \frac{d^k}{dt^k} \log(1+\frac{\tau_{m-1}(t+1)}{\phi(t+1)})\\$ And $||...||_{a\le t \le b}$ is the sup-norm across some interval $[a,b] \subset (-2,\infty)$. As a forewarning, this is going to be very messy... Now to begin we can bound, $ ||\phi^{(j)}(t)||_{a\le t \le b} \le M\,\,\text{for}\,\,j\le k\\$ And that next, $ \phi^{(k)}(t+1) + \tau_m^{(k)}(t+1) = \frac{d^k}{dt^k} e^{\phi(t) +\tau_{m+1}(t)}\\ = \sum_{j=0}^k \binom{k}{j} (\frac{d^{k-j}}{dt^{k-j}} \phi(t+1)e^{-t})(\frac{d^j}{dt^j} e^{\tau_{m+1}(t)})\\ = \sum_{j=0}^{k-1} \binom{k}{j} (\frac{d^{k-j}}{dt^{k-j}} \phi(t+1)e^{-t})(\frac{d^j}{dt^j} e^{\tau_{m+1}(t)}) + \phi(t+1)e^{-t}(\frac{d^k}{dt^k} e^{\tau_{m+1}(t)})$ Now, $ \frac{d^k}{dt^k} e^{\tau_{m+1}(t)} = e^{\tau_{m+1}(t)}(\tau_{m+1}^{(k)}(t) + \sum_{j=0}^{k-1} a_j \tau_{m+1}^{(j)}(t))\\$ So, we ask you to put on your thinking cap, and excuse me if I write, $ \tau_{m}^{(k)}(t+1) = A_m + C_m\tau_{m+1}^{(k)}(t)\\$ And by the induction hypothesis, $ \sum_{j=1}^\infty ||A_{j+1} - A_j||_{a\le t \le b} < \infty\\ \sum_{j=1}^\infty ||C_{j+1} - C_j||_{a\le t \le b} < \infty\\$ Which is because these terms are made up of finite sums and products of $\tau_m^{(j)}$ and these are said to be summable.  Now the proof is a walk in the park. $ \tau_{m}^{(k)}(t) = \frac{\tau_{m-1}^{(k)}(t+1) - A_{m-1}}{C_{m-1}}... = \sum_{j=0}^{m-1} (\prod_{k=0}^{m-1-j} C_{m-1-k}^{-1}) A_j\\$ Where, we've continued the iteration and set $\tau_0 = 0$ and $\tau_1 = 0$ for $k>1$, and $\tau_1 = 1$ for $k=1$ (but we're tossing this away because we know it's differentiable). Therefore, $ \sum_{m=1}^\infty ||\tau_{m+1}^{(k)}(t) - \tau_m^{(k)}(t)||_{a \le t \le b} < \infty\\$ 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 $C^\infty$ out of the way, we ask if we can continue this iteration and get pentation.  Now, $\text{slog}$ will certainly be $C^{\infty}$ and $\frac{d}{dt}e \uparrow \uparrow t > 0$ so it's a well defined bijection of $\mathbb{R} \to (-2,\infty)$. So, first up to bat is to get another phi function, $ \Phi(t) = \Omega_{j=1}^\infty e^{t-j} e \uparrow \uparrow x \bullet x = e^{t-1} e \uparrow \uparrow (e^{t-2}e \uparrow \uparrow (e^{t-3} e \uparrow \uparrow ...)) $ This will be $C^\infty$ (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, $ \Phi(t+1) = e^t (e \uparrow \uparrow \Phi(t))\\$ By now, I think you might know where i'm going with this. $ e \uparrow^3 t = \lim_{n\to\infty} \text{slog} \text{slog} \cdots (n\,\text{times})\cdots\text{slog} \Phi(t+\omega_1 + n)\\$ 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... « Next Oldest | Next Newest »

 Messages In This Thread On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/08/2021, 12:12 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/10/2021, 02:30 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 02/10/2021, 04:09 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/10/2021, 08:10 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/16/2021, 08:40 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 02/21/2021, 01:38 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by tommy1729 - 02/27/2021, 12:08 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 02/27/2021, 09:57 PM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by MphLee - 03/01/2021, 11:22 PM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 03/02/2021, 05:09 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by MphLee - 02/27/2021, 11:37 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 03/02/2021, 09:55 PM

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