# Tetration Forum

Full Version: Complex Tetration, to base exp(1/e)
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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
(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.
(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.
(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    */
[attachment=1344]
(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.
(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).
Sheldon, I am glad you helped out on this question, I am - like always - in limited time mode.
(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."