• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 On extension to "other" iteration roots Leo.W Junior Fellow Posts: 30 Threads: 3 Joined: Apr 2021 09/28/2021, 12:46 PM (This post was last modified: 09/28/2021, 12:53 PM by Leo.W.) (09/26/2021, 10:53 PM)JmsNxn Wrote: (09/25/2021, 01:49 PM)Leo.W Wrote: (09/25/2021, 02:59 AM)JmsNxn Wrote: ... Regards, JamesThank you, James! ... Leo Does the P approach generalize to other functions? I will admit I'm still a little confused by it, but it seems to be working, lol. I ask because I don't see anything too specific to tetration, so I wonder if it works in more elaborate scenarios. regards, JamesOk, so I generate this like 5 minutes ago, it shows P approach is quite applicable, as I said in update 1, I would generate a tetration based sqrt(2) and oscillating between two branch of its superfunctions, the first one is T1(0)=1, the second one is T2(0)=3, P approach will merge T1 and T2 into W in a way that $W(0)=1,W(\frac{1}{2})=3,W(z+1)=\sqrt{2}^{W(z)}$ for all z except for some branch cuts. This is the value table and plot of such W: Code:{0.0000000000, 1.000000000}, {0.0100000000, 1.013941867}, {0.0200000000, 1.043390887}, {0.0300000000, 1.087028199}, {0.0400000000, 1.143082118}, {0.0500000000, 1.209493951}, {0.0600000000, 1.284085589}, {0.0700000000, 1.364706745}, {0.0800000000, 1.449348480}, {0.0900000000, 1.536218653}, {0.1000000000, 1.623781865}, {0.1100000000, 1.710770523}, {0.1200000000, 1.796175081}, {0.1300000000, 1.879221094}, {0.1400000000, 1.959339408}, {0.1500000000, 2.036134108}, {0.1600000000, 2.109351304}, {0.1700000000, 2.178850589}, {0.1800000000, 2.244580027}, {0.1900000000, 2.306554960}, {0.2000000000, 2.364840481}, {0.2100000000, 2.419537251}, {0.2200000000, 2.470770225}, {0.2300000000, 2.518679839}, {0.2400000000, 2.563415217}, {0.2500000000, 2.605129041}, {0.2600000000, 2.643973721}, {0.2700000000, 2.680098616}, {0.2800000000, 2.713648057}, {0.2900000000, 2.744759999}, {0.3000000000, 2.773565147}, {0.3100000000, 2.800186444}, {0.3200000000, 2.824738818}, {0.3300000000, 2.847329133}, {0.3400000000, 2.868056270}, {0.3500000000, 2.887011312}, {0.3600000000, 2.904277785}, {0.3700000000, 2.919931943}, {0.3800000000, 2.934043071}, {0.3900000000, 2.946673798}, {0.4000000000, 2.957880399}, {0.4100000000, 2.967713097}, {0.4200000000, 2.976216345}, {0.4300000000, 2.983429084}, {0.4400000000, 2.989384991}, {0.4500000000, 2.994112694}, {0.4600000000, 2.997635975}, {0.4700000000, 2.999973941}, {0.4800000000, 3.001141180}, {0.4900000000, 3.001147892}, {0.5000000000, 3.000000000}, {0.5100000000, 2.997699243}, {0.5200000000, 2.994243251}, {0.5300000000, 2.989625606}, {0.5400000000, 2.983835882}, {0.5500000000, 2.976859684}, {0.5600000000, 2.968678666}, {0.5700000000, 2.959270557}, {0.5800000000, 2.948609168}, {0.5900000000, 2.936664416}, {0.6000000000, 2.923402349}, {0.6100000000, 2.908785182}, {0.6200000000, 2.892771364}, {0.6300000000, 2.875315665}, {0.6400000000, 2.856369317}, {0.6500000000, 2.835880203}, {0.6600000000, 2.813793125}, {0.6700000000, 2.790050161}, {0.6800000000, 2.764591146}, {0.6900000000, 2.737354289}, {0.7000000000, 2.708276982}, {0.7100000000, 2.677296816}, {0.7200000000, 2.644352863}, {0.7300000000, 2.609387284}, {0.7400000000, 2.572347293}, {0.7500000000, 2.533187568}, {0.7600000000, 2.491873166}, {0.7700000000, 2.448383000}, {0.7800000000, 2.402713965}, {0.7900000000, 2.354885746}, {0.8000000000, 2.304946356}, {0.8100000000, 2.252978407}, {0.8200000000, 2.199106047}, {0.8300000000, 2.143502472}, {0.8400000000, 2.086397761}, {0.8500000000, 2.028086706}, {0.8600000000, 1.968936132}, {0.8700000000, 1.909391018}, {0.8800000000, 1.849978583}, {0.8900000000, 1.791309293}, {0.9000000000, 1.734073681}, {0.9100000000, 1.679033859}, {0.9200000000, 1.627008772}, {0.9300000000, 1.578852631}, {0.9400000000, 1.535426592}, {0.9500000000, 1.497564563}, {0.9600000000, 1.466035029}, {0.9700000000, 1.441501698}, {0.9800000000, 1.424486490}, {0.9900000000, 1.415338668}, {1.0000000000, 1.414213562} But as I updated statements, if two of these superfunctions have a different limit at the same infinity, they can't be merged, otherwise they're almost always merge-able. This statement can be proved, I may explain it later. Also it shows you can never generate a real-to-real tetration for base 0=e^-e. I'll be working on bases 0= 0] := Nest[Log[Sqrt[2], SetPrecision[#, 2000]] &,  Q2[SetPrecision[    z + 100 -     0.003512881867631336045661822591552871843014614532697105512130826\ 7177706904795348344145904176580310159711371579929476100., 2000]], 100] Q2R[z_ /; Re[z] < 0] := Log[Sqrt[2], Q2R[z + 1]] P[z_] := 1/2 + Cos[Pi z]/2 W[z_] := P[2 z] Q1R[z] + P[2 z + 1] Q2R[z - 1/2] WW[z_] := SetPrecision[  Nest[Log[Sqrt[2], SetPrecision[#, 200]] &, W[z + 100], 100], 100] Leo Attached Files Image(s) « Next Oldest | Next Newest »

 Messages In This Thread On extension to "other" iteration roots - by Leo.W - 09/24/2021, 04:25 PM RE: On extension to "other" iteration roots - by JmsNxn - 09/25/2021, 02:59 AM RE: On extension to "other" iteration roots - by Leo.W - 09/25/2021, 01:49 PM RE: On extension to "other" iteration roots - by JmsNxn - 09/26/2021, 10:53 PM RE: On extension to "other" iteration roots - by Leo.W - 09/28/2021, 12:46 PM RE: On extension to "other" iteration roots - by Leo.W - 09/28/2021, 02:24 PM RE: On extension to "other" iteration roots - by JmsNxn - 09/29/2021, 12:46 AM RE: On extension to "other" iteration roots - by Leo.W - 09/29/2021, 04:12 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Ueda - Extension of tetration to real and complex heights MphLee 2 388 12/03/2021, 01:23 AM Last Post: JmsNxn Possible continuous extension of tetration to the reals Dasedes 0 2,861 10/10/2016, 04:57 AM Last Post: Dasedes Solving tetration using differintegrals and super-roots JmsNxn 0 3,641 08/22/2016, 10:07 PM Last Post: JmsNxn Andrew Robbins' Tetration Extension bo198214 32 71,984 08/22/2016, 04:19 PM Last Post: Gottfried Non-trivial extension of max(n,1)-1 to the reals and its iteration. MphLee 3 7,181 05/17/2014, 07:10 PM Last Post: MphLee extension of the Ackermann function to operators less than addition JmsNxn 2 6,904 11/06/2011, 08:06 PM Last Post: JmsNxn Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 4,604 09/04/2011, 05:59 AM Last Post: Gottfried Fractional iteration of x^2+1 at infinity and fractional iteration of exp bo198214 10 26,827 06/09/2011, 05:56 AM Last Post: bo198214 Pentation roots self but please you do... nuninho1980 2 10,294 11/03/2010, 12:54 PM Last Post: nuninho1980 proof: Limit of self-super-roots is e^1/e. TPID 6 bo198214 3 11,083 07/10/2010, 09:13 AM Last Post: bo198214

Users browsing this thread: 1 Guest(s)