Thread Rating:
  • 3 Vote(s) - 4.33 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Searching for an asymptotic to exp[0.5]
#6
(05/10/2014, 12:14 PM)sheldonison Wrote: One obvious questions from the Taylor series result, that I can't answer, because I have no idea how fast these functions grow as x goes to infinity, relative to exponentiaton. What is the "growth" of a functions like these, which should grow slower than exponentiation, but faster than any polynomial?


Wow -- both of these functions grow exactly exponentially, where growth is defined as slog(f^n)/n.

The first function is


But If my math is correct then iterating f is the same as iterating , which is ..... the same as iterating an exponential!!!

The second function is


Ok, the second function is a little more complicated, but if my math is correct, it is going to be the same as iterating , which .... drumroll ... is also exponential growth!

So, half exponential functions, especially the Taylor series of entire versions of half exponential functions need more study... Lets conjecture that I have a constructive definition of an entire half exponential Taylor series, for which I haven't given all of the details, but I have a pari-gp program. Then the Taylor series coefficients must eventually grow slower than all of these entire functions with exponential growth.... really interesting!!!
- Sheldon


Reply


Messages In This Thread
RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/10/2014, 11:48 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Using a family of asymptotic tetration functions... JmsNxn 15 4,597 08/06/2021, 01:47 AM
Last Post: JmsNxn
  Reducing beta tetration to an asymptotic series, and a pull back JmsNxn 2 883 07/22/2021, 03:37 AM
Last Post: JmsNxn
  A Holomorphic Function Asymptotic to Tetration JmsNxn 2 1,327 03/24/2021, 09:58 PM
Last Post: JmsNxn
  An asymptotic expansion for \phi JmsNxn 1 1,150 02/08/2021, 12:25 AM
Last Post: JmsNxn
  Merged fixpoints of 2 iterates ? Asymptotic ? [2019] tommy1729 1 3,548 09/10/2019, 11:28 AM
Last Post: sheldonison
  Another asymptotic development, similar to 2sinh method JmsNxn 0 4,091 07/05/2011, 06:34 PM
Last Post: JmsNxn



Users browsing this thread: 1 Guest(s)