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 Tetra-series Gottfried Ultimate Fellow Posts: 770 Threads: 120 Joined: Aug 2007 06/13/2008, 07:15 AM (This post was last modified: 06/13/2008, 07:34 AM by Gottfried.) Assume for the following a fixed base t for the dexp- or "U"-tetration, so that dxp°h(x):=dxp_t°h(x) and for shortness let me replace dxp by U in the following. Also I change some namings from the previous posts for consistency. Denote the alternating series of U-powertowers of increasing positive heights $\hspace{24} asp(x) =\sum_{h=0}^{\infty} (-1)^h U^{\circ h}(x)$ and of increasing negative heights $\hspace{24} asn(x) =\sum_{h=0}^{\infty} (-1)^h U^{\circ -h}(x)$ then my conjecture, based on diagonalization was $\hspace{24} asp(x) + asn(x) - x =0$ which was wrong with a certain systematic error $\hspace{24} asp(x) + asn(x) - x =d(x)$ I have now a description for the error d(x), which fits very well. First, let's formally write $\hspace8 as(x) = asp(x)+asn(x)-x$as the two-way-infinite series $\hspace{24} as(x) =\sum_{h=-\infty}^{\infty} (-1)^h*U^{\circ h}(x)$ Then it is obvious, that as(x) is periodic with 2h, if x is expressed as powertower $\hspace{24} x =U^{\circ y}(1) = U^{\circ 2h+r}(1)$ where h is integer and r is the fractional remainder of a number y (mod 2) $\hspace{24} x= U^{\circ 2h+r}(1)$ and define $\hspace{24} 1_r= U^{\circ r}(1)$ Then we may discuss as_r = as(x) as $\hspace{24} as_r = as(U^{\circ r}(1))=as(1_r)$ where r = y (mod 2) We can then rewrite the formula $\hspace{24} asp(x)+asn(x)-x = d(x)$ as $\hspace{24} asp_r + asn_r - 1_r = as_r = d_r$ My observation was, that d_r is sinusoidal with r, with a very good fit (I checked also various bases t). I got now $\hspace{24} as_r (=d_r) = a * \sin(r *\pi + w)$ where "a" indicates the amplitude and "w" a phase-shift (depending on base t). The following fits the result very well: $\hspace{24} a = \sqrt{ (as_0)^2 + (as_{0.5})^2)}$ $\hspace{24} w = \text{atan2}(as_0 , as_{0.5})$ so we could rewrite $\hspace{24} asp_r + asn_r - 1_r$ as functional equation for asp $\hspace{24} asp_r = a * \sin(r*\pi + w) - (asn_r - 1_r)$ and also can determine all as_r using asn_r for r=0 and r=0.5 (the integer and half-integer-iterate U°0(1) and U°0.5(1)) only. Note, that the computation of asp_r is exact using the appropriate matrix (I + Ut)^-1 of the diagonalization-method. The diagonalization-method deviates only for the part asn_r; it gives asn_r - d_r instead of asn_r ; unfortunately, this does not allow to determine d_r correctly with the diagonalization-method only (yet). The benefit of the diagonalization-method is here, that its matrix (I+Ut)^-1 provides the coefficients for a powerseries for asp_r, which seems to be the analytic continuation for bases t, where the series asp would diverge. Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Tetra-series - by Gottfried - 11/20/2007, 12:47 PM RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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