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Complex Tetration, to base exp(1/e)
#1
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
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#2
(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 which is congruent to iterating
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 , 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.  

If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.  

To get arbitrarily accurate results, we iterate enough times or for the repellilng flower, we can iterate enough times so that z is small and the asymptotic series works well. 
- Sheldon
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#3
(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 which is congruent to iterating
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 , 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.  

If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.  

To get arbitrarily accurate results, we iterate enough times or for the repellilng flower, we can iterate 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.
Reply
#4
(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    */

.gp   baseeta.gp (Size: 6.4 KB / Downloads: 19)
- Sheldon
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#5
(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.
Reply


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