Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
closed form for regular superfunction expressed as a periodic function
(08/28/2010, 11:21 PM)tommy1729 Wrote: that seems efficient and intresting.

in fact i doubt it hasnt been considered before ?
Thanks Tommy! I assume it has been considered, and probably calculated before. I think Kneser developed the complex periodic regular tetration for base e, and probably would've generated the coefficients. But I haven't seen them before. Perhaps Henryk (or someone else) could comment???

I figured out the closed form equation for a couple more terms, and I have an equation that should generate the other terms, but I'm still working it, literally as I write this post!

What I did is start with the equation:

and set it equal to the equation

Continuing, there is a bit of trickery in this step to keep the equations in terms of , instead of in terms of . Notice that .

This becomes a product, with and

The goal is to get an equation in terms of on both sides of the equation. Then I had a breakthrough, while I was typing this post!!!! The breakthrough is to set , and rewrite all of the equations in terms of y! This wraps the 2Pi*I/L cyclic Fourier series around the unit circle, as an analytic function in terms of y, which greatly simplifies the equations, and also helps to justify the equations.

The next step is to expand the individual Tayler series for the , and multiply them all together (which gets a little messy, but remember a0=L and a1=1), and finally equate the terms in on the left hand side equation with those on the right hand side equation, and solve for the individual coefficients. Anyway, the equations match the numerical results.

I'll fill in the Tayler series substitution next time; this post is already much more detailed then I thought it was going to be! I figured a lot of this out as I typed this post!
- Sheldon


Messages In This Thread
RE: regular superfunction expressed as a periodic function - by sheldonison - 08/30/2010, 03:09 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  New mathematical object - hyperanalytic function arybnikov 4 1,244 01/02/2020, 01:38 AM
Last Post: arybnikov
  Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 668 09/09/2019, 10:55 PM
Last Post: tommy1729
  Is there a function space for tetration? Chenjesu 0 722 06/23/2019, 08:24 PM
Last Post: Chenjesu
  Degamma function Xorter 0 1,178 10/22/2018, 11:29 AM
Last Post: Xorter
  Periodic analytic iterations by Riemann mapping tommy1729 1 2,577 03/05/2016, 10:07 PM
Last Post: tommy1729
  Should tetration be a multivalued function? marraco 17 19,207 01/14/2016, 04:24 AM
Last Post: marraco
  Introducing new special function : Lambert_t(z,r) tommy1729 2 4,131 01/10/2016, 06:14 PM
Last Post: tommy1729
  Natural cyclic superfunction tommy1729 3 3,502 12/08/2015, 12:09 AM
Last Post: tommy1729
Sad Tommy-Mandelbrot function tommy1729 0 2,226 04/21/2015, 01:02 PM
Last Post: tommy1729
  Can sexp(z) be periodic ?? tommy1729 2 4,309 01/14/2015, 01:19 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)