• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 closed form for regular superfunction expressed as a periodic function sheldonison Long Time Fellow Posts: 633 Threads: 22 Joined: Oct 2008 08/30/2010, 03:09 AM (This post was last modified: 08/30/2010, 03:33 AM by sheldonison.) (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! $a_2 = (1/2)/(L - 1)$ $a_3 = (1/6 + a_2)/(L*L - 1)$ $a_4 = (1/24 + (1/2)*a_2*a_2 + (1/2)*a_2 + a_3)/(L*L*L-1)$ What I did is start with the equation: $\text{RegularSuperf}(z) = \sum_{n=0}^{\infty}a_nL^{nz}$ and set it equal to the equation $\text{RegularSuperf}(z) = \exp{(\text{RegularSuperf}(z-1))}$ Continuing, there is a bit of trickery in this step to keep the equations in terms of $L^{nz}$, instead of in terms of $L^{n(z-1)}$. Notice that $L^{n(z-1)}=L^{(nz-n)}=L^{-n}L^{nz}$. $\text{RegularSuperf}(z) = \exp{(\text{RegularSuperf}(z-1))} = \exp{( \sum_{n=0}^{\infty}\exp^{(L^{-n}a_nL^{nz})})}$ This becomes a product, with $a_0=L$ and $a_1=1$ $\text{RegularSuperf}(z) = \prod_{n=0}^{\infty} \exp{(L^{-n}a_nL^{nz})}$ The goal is to get an equation in terms of $L^{nz}$ on both sides of the equation. Then I had a breakthrough, while I was typing this post!!!! The breakthrough is to set $y=L^z$, 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. $\text{RegularSuperf}(z) = \sum_{n=0}^{\infty}a_ny^n = \prod_{n=0}^{\infty} \exp{(L^{-n}a_ny^n)}$ The next step is to expand the individual Tayler series for the $\exp {(L^{-n}a_ny^n)}$, and multiply them all together (which gets a little messy, but remember a0=L and a1=1), and finally equate the terms in $y^n$ on the left hand side equation with those on the right hand side equation, and solve for the individual $a_n$ 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 « Next Oldest | Next Newest »

 Messages In This Thread closed form for regular superfunction expressed as a periodic function - by sheldonison - 08/27/2010, 02:09 PM RE: regular superfunction expressed as a periodic function - by sheldonison - 08/28/2010, 08:44 PM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/28/2010, 11:21 PM RE: regular superfunction expressed as a periodic function - by sheldonison - 08/30/2010, 03:09 AM RE: regular superfunction expressed as a periodic function - by Gottfried - 08/30/2010, 09:22 AM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/30/2010, 09:41 AM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/30/2010, 09:46 AM RE: regular superfunction expressed as a periodic function - by Gottfried - 08/31/2010, 08:34 PM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/03/2010, 08:19 PM RE: closed form for regular superfunction expressed as a periodic function - by sheldonison - 09/05/2010, 05:36 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/05/2010, 04:45 PM RE: closed form for regular superfunction expressed as a periodic function - by sheldonison - 09/07/2010, 03:54 PM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/07/2010, 07:46 PM RE: closed form for regular superfunction expressed as a periodic function - by bo198214 - 09/08/2010, 06:03 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/08/2010, 06:55 PM RE: closed form for regular superfunction expressed as a periodic function - by bo198214 - 09/09/2010, 10:12 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/09/2010, 10:18 PM

 Possibly Related Threads... Thread Author Replies Views Last Post New mathematical object - hyperanalytic function arybnikov 4 223 01/02/2020, 01:38 AM Last Post: arybnikov Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 346 09/09/2019, 10:55 PM Last Post: tommy1729 Is there a function space for tetration? Chenjesu 0 457 06/23/2019, 08:24 PM Last Post: Chenjesu Degamma function Xorter 0 903 10/22/2018, 11:29 AM Last Post: Xorter Periodic analytic iterations by Riemann mapping tommy1729 1 2,253 03/05/2016, 10:07 PM Last Post: tommy1729 Should tetration be a multivalued function? marraco 17 16,911 01/14/2016, 04:24 AM Last Post: marraco Introducing new special function : Lambert_t(z,r) tommy1729 2 3,661 01/10/2016, 06:14 PM Last Post: tommy1729 Natural cyclic superfunction tommy1729 3 3,064 12/08/2015, 12:09 AM Last Post: tommy1729 Tommy-Mandelbrot function tommy1729 0 1,981 04/21/2015, 01:02 PM Last Post: tommy1729 Can sexp(z) be periodic ?? tommy1729 2 3,880 01/14/2015, 01:19 PM Last Post: tommy1729

Users browsing this thread: 2 Guest(s)