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 On to C^\infty--and attempts at C^\infty hyper-operations sheldonison Long Time Fellow Posts: 679 Threads: 24 Joined: Oct 2008 02/21/2021, 01:38 AM (02/16/2021, 08:40 AM)JmsNxn Wrote: So I posted a proof of $C^\infty$ before I started working today... I am in the process of rewriting my entire paper to focus on $C^\infty$ hyper-operations; whereby it's a long proof by induction. But the initial step is to prove that this tetration is $C^\infty$. Now I can most definitely show this tetration is $C^\infty$; the trouble I'm having is making the proof as general as possible; so that we can create a proof by induction showing $e \uparrow^k t$ is $C^\infty$. Hi James,   I like your paper.  I would suggest generating an infinite sequence of entire $\phi_n$ functions, perhaps defined as follows; this is slightly modified from your approach where this $\phi_2(s)$ = JmsNxn phi(s+1) we could start with  $\phi_1(s)=\exp(s)$ $\phi_2(s)=\exp(\phi_2(s-1)+s);\;$ this $\phi_2(s)$ asymptotically approaches exp(s) as $\Re(s)$ gets arbitrarily negative,  $\phi_n(s)=\phi_{n-1}(\phi_{n}(s-1)+s);\;$ $\phi_n(s)$ also asymptotically approaches exp(s) as $\Re(s)$ gets arbitrarily negative James has proven that $\phi_2(s)$ is entire, and I think each of these phi functions is also entire, and each $\phi_n(s)$ would probably lead to an $e\uparrow^n(s)\;$ function which is also $C^\infty$ only defined at the real axis; details tbd... - Sheldon « 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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