Complex Tetration, to base exp(1/e) Ember Edison Fellow Posts: 90 Threads: 10 Joined: May 2019 05/05/2019, 11:38 PM Hi, I was reading the article[1] and i can't reproduce it in mathematica. I need some help, and very much need some code. Edison [1]https://arxiv.org/abs/1105.4735 sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 05/07/2019, 04:17 PM (This post was last modified: 05/07/2019, 04:46 PM by sheldonison.) (05/05/2019, 11:38 PM)Ember Edison Wrote: Hi, I was reading the article[1] and i can't reproduce it in mathematica. I need some help, and very much need some code. Edison [1]https://arxiv.org/abs/1105.4735 Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating $z\mapsto\exp(z)-1$ which is congruent to iterating $\eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}-1;$ The asymptotic series for the Abel equation for iterating z is given by equation 18.  I have used this equation to also get the value of Tetration or superfunction for base $\eta=\exp(1/e)$, by using a good initial estimate, and then Newton's method.  If you use pari-gp or are interested in downloading pari-gp, I can post the pari-gp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.   $\alpha(z)=-\frac{2}{z}+\frac{1}{3}\log(\pm z)-\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3-\frac{71}{435456}z^4+...$ If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.   $\alpha(z)\approx\alpha(\exp(z)-1)+1$ To get arbitrarily accurate results, we iterate $z\mapsto\exp(z)-1$ enough times or for the repellilng flower, we can iterate $z\mapsto\log(z+1)$ enough times so that z is small and the asymptotic series works well.  - Sheldon Ember Edison Fellow Posts: 90 Threads: 10 Joined: May 2019 05/08/2019, 12:25 PM (05/07/2019, 04:17 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Hi, I was reading the article[1] and i can't reproduce it in mathematica. I need some help, and very much need some code. Edison [1]https://arxiv.org/abs/1105.4735 Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating $z\mapsto\exp(z)-1$ which is congruent to iterating $\eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}-1;$ The asymptotic series for the Abel equation for iterating z is given by equation 18.  I have used this equation to also get the value of Tetration or superfunction for base $\eta=\exp(1/e)$, by using a good initial estimate, and then Newton's method.  If you use pari-gp or are interested in downloading pari-gp, I can post the pari-gp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.   $\alpha(z)=-\frac{2}{z}+\frac{1}{3}\log(\pm z)-\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3-\frac{71}{435456}z^4+...$ If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.   $\alpha(z)\approx\alpha(\exp(z)-1)+1$ To get arbitrarily accurate results, we iterate $z\mapsto\exp(z)-1$ enough times or for the repellilng flower, we can iterate $z\mapsto\log(z+1)$ enough times so that z is small and the asymptotic series works well. Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful. sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 05/08/2019, 04:50 PM (This post was last modified: 05/08/2019, 05:38 PM by sheldonison.) (05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful.[attachment=1343] Code:\r baseeta.gp initeta();         /* initeta initializes kecalle series; 25terms */ slog1=slogeta(1);  /* renormalize so slog(1)=0; slog1=3.029297214418036; */ z=slogeta(2.5)     /* 21.038456088895745460253062718325504556;    */ ploth(t=-1.5,25,sexpeta(t));  /* plot of sexpeta; sexpeta(0)=1    */ z2=invcheta(100)   /*  0.79336896191958487417879655443666434028   */ z1=invcheta(4);    /* -4.5049005907984782975089673142337641018    */ ploth(t=z1,z2,cheta(t));  /* plot of upper superfucntion of eta   */ z=slogeta(I)       /* -1.217279555798763 + 0.5193692007946583*I   */ z=invcheta(I)      /*  1.808671078843811 + 1.565868985090261*I    */ z=cheta(1+I)       /* -6.501975132474055 + 4.920389603877520*I    */   baseeta.gp (Size: 6.4 KB / Downloads: 543) - Sheldon Ember Edison Fellow Posts: 90 Threads: 10 Joined: May 2019 05/08/2019, 06:20 PM (05/08/2019, 04:50 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful. Code:\r baseeta.gp initeta();         /* initeta initializes kecalle series; 25terms */ slog1=slogeta(1);  /* renormalize so slog(1)=0; slog1=3.029297214418036; */ z=slogeta(2.5)     /* 21.038456088895745460253062718325504556;    */ ploth(t=-1.5,25,sexpeta(t));  /* plot of sexpeta; sexpeta(0)=1    */ z2=invcheta(100)   /*  0.79336896191958487417879655443666434028   */ z1=invcheta(4);    /* -4.5049005907984782975089673142337641018    */ ploth(t=z1,z2,cheta(t));  /* plot of upper superfucntion of eta   */ z=slogeta(I)       /* -1.217279555798763 + 0.5193692007946583*I   */ z=invcheta(I)      /*  1.808671078843811 + 1.565868985090261*I    */ z=cheta(1+I)       /* -6.501975132474055 + 4.920389603877520*I    */ Thank you! I am reading. Ember Edison Fellow Posts: 90 Threads: 10 Joined: May 2019 08/06/2019, 05:22 PM (05/08/2019, 04:50 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful. Code:\r baseeta.gp initeta();         /* initeta initializes kecalle series; 25terms */ slog1=slogeta(1);  /* renormalize so slog(1)=0; slog1=3.029297214418036; */ z=slogeta(2.5)     /* 21.038456088895745460253062718325504556;    */ ploth(t=-1.5,25,sexpeta(t));  /* plot of sexpeta; sexpeta(0)=1    */ z2=invcheta(100)   /*  0.79336896191958487417879655443666434028   */ z1=invcheta(4);    /* -4.5049005907984782975089673142337641018    */ ploth(t=z1,z2,cheta(t));  /* plot of upper superfucntion of eta   */ z=slogeta(I)       /* -1.217279555798763 + 0.5193692007946583*I   */ z=invcheta(I)      /*  1.808671078843811 + 1.565868985090261*I    */ z=cheta(1+I)       /* -6.501975132474055 + 4.920389603877520*I    */ Sorry, I think we need penteta, ipenteta, hexeta, ihexeta in fatou.gp because pentinit(etaB) is use sexpinit(etaB). bo198214 Administrator Posts: 1,624 Threads: 103 Joined: Aug 2007 08/13/2019, 08:27 PM Sheldon, I am glad you helped out on this question, I am - like always - in limited time mode. sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 08/14/2019, 09:15 AM (08/13/2019, 08:27 PM)bo198214 Wrote: Sheldon, I am glad you helped out on this question, I am - like always - in limited time mode. Thanks you for your kind comments Henryk.  It has been a pleasure to learn more and more about the start of the art of complex dynamics.  I still don't quite understand all of Shishikura's papers, "Bifurcation of parabolic fixed points", an in particular, how Shishikura used perturbed fatou coordinates in his other proofs.  "In fact, in [Sh1], such a notion was already introduced and its second iterate played a crucial role in the proof of the fact that a parabolic point can be perturbed so that the Hausdorff dimension of the Julia set is arbitrarily close to 2." - Sheldon « Next Oldest | Next Newest »

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