Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
on constructing hyper operations for bases > eta
Hello everyone. So as I put in my last thread here, I have a solution to hyper operations for bases between 1 and . The natural question is if we can extend our results. As fivexthethird pointed out we can apply these ideas to the super root function. Then as Tommy pointed out this can then again result in a provable recursion.

To be clean and precise, I will explain the methodology here.

and if converges and exists then quite exactly since is the inverse of it is calcuable for and then

Inverting this function gives us a solution to tetration, proving the recursion with a little breath of thought. The problem is proving convergence of the differintegral. If we can do this we can iterate the procedure and solve for pentation then hexation and then septation, etc... This will give us a solution to hyper operations with a base on the positive real line. Not only that, an incredibly economical solution.

I've begun work on this and I think the procedure for proving convergence is the only problem, and then I think I got the knack of the result. I'm performing a similar induction schema and I think the result only needs a little bit of analysis magic that I've yet to find. But I'm sure it's out there.

I'm posting this thread to see if anyone has any ideas. I'm very excited about this result and it tittilates me to see it proved. I am wanting to collaborate on a paper with whomever wants to work on these ideas. I think some of the steps may be more complicated than first expected. I'm sort of starting with the idea of convergence, as I am sure that is the hardest hill to climb.

Thanks again, and hope you guys can help.
If you use it to calculate , how does it compares with with Cosh(x)?

I suspect that both functions are equal, (or really, really close), at least on a radius around the origin.

Possibly Related Threads…
Thread Author Replies Views Last Post
Question Rank-Wise Approximations of Hyper-Operations Catullus 43 1,985 07/01/2022, 10:01 PM
Last Post: tommy1729
  Thoughts on hyper-operations of rational but non-integer orders? VSO 4 4,375 06/30/2022, 11:41 PM
Last Post: MphLee
Question Hyper-Operations, Base Pi Catullus 2 123 06/24/2022, 08:46 AM
Last Post: Catullus
Question Weak Hyper-Operational etas and Euler Numbers Catullus 0 79 06/17/2022, 09:45 AM
Last Post: Catullus
Question Octonion Hyper-Operations Catullus 2 168 06/12/2022, 02:12 PM
Last Post: tommy1729
  Tetration and higher-order operations on transfinite ordinals quickfur 9 20,202 06/09/2022, 10:50 PM
Last Post: Catullus
  On my old fractional calculus approach to hyper-operations JmsNxn 14 5,472 07/07/2021, 07:35 AM
Last Post: JmsNxn
  hyper 0 dantheman163 2 5,914 03/09/2021, 10:28 PM
Last Post: MphLee
  On to C^\infty--and attempts at C^\infty hyper-operations JmsNxn 11 5,630 03/02/2021, 09:55 PM
Last Post: JmsNxn
  Could there be an "arctic geometry" by raising the rank of all operations? Syzithryx 2 4,744 07/24/2019, 05:59 PM
Last Post: Syzithryx

Users browsing this thread: 1 Guest(s)