Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
closed form for regular superfunction expressed as a periodic function
#12
(08/31/2010, 07:08 AM)Gottfried Wrote: Hi Sheldon -

I recognize your coefficients....
Gottfried, thanks for your reply. I'm trying to make sense of the coefficients, and would probably need to program the matrix into pari-gp to verify it.

Earlier, Gottfried wrote:
Quote:Hi Sheldon -

just to allow me to follow (think I can't involve much) - I don't have a clue from where this is coming, what, for instance, is L at all? I think you've explained it elsewhere before but don't see it at the moment... Would you mind to reexplain in short or to provide the link?

Gottfried
L is the fixed point of base(e). If you iterate the natural logarithm function hundreds of times (say starting with z=0.5), you get a very good approximation of L. Then the equations describe the regular superfunction for base e, which is complex valued at the real axis, but is analytic and entire.

The rest of what I did basically, it comes down to expressing the regular superfunction as an analytic Fourier series, where all of the terms of the series decay to zero at i*infinity. Such a series is guaranteed to be analytic.

Here is an example of a generalized view of such a Fourier series, not related to the superfunction, with a period of 2Pi.

A full Fourier series would also have the negative coefficients as well, and is often not analytic.

Both of these can be wrapped around the unit circle, with the substitution . In the analytic case, we have an analytic Taylor series. . In the general case, with terms from -infinty to +infinty, we have a Laurent series, with singularities inside the unit circle, and an annular ring of convergence. For the full Fourier series, often, the Laurent series only converges on the edge of the unit circle. Hope that helps.
- Sheldon



Reply


Messages In This Thread

Possibly Related Threads...
Thread Author Replies Views Last Post
  New mathematical object - hyperanalytic function arybnikov 4 1,065 01/02/2020, 01:38 AM
Last Post: arybnikov
  Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 619 09/09/2019, 10:55 PM
Last Post: tommy1729
  Is there a function space for tetration? Chenjesu 0 674 06/23/2019, 08:24 PM
Last Post: Chenjesu
  Degamma function Xorter 0 1,128 10/22/2018, 11:29 AM
Last Post: Xorter
  Periodic analytic iterations by Riemann mapping tommy1729 1 2,514 03/05/2016, 10:07 PM
Last Post: tommy1729
  Should tetration be a multivalued function? marraco 17 18,697 01/14/2016, 04:24 AM
Last Post: marraco
  Introducing new special function : Lambert_t(z,r) tommy1729 2 4,020 01/10/2016, 06:14 PM
Last Post: tommy1729
  Natural cyclic superfunction tommy1729 3 3,402 12/08/2015, 12:09 AM
Last Post: tommy1729
Sad Tommy-Mandelbrot function tommy1729 0 2,182 04/21/2015, 01:02 PM
Last Post: tommy1729
  Can sexp(z) be periodic ?? tommy1729 2 4,205 01/14/2015, 01:19 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)