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Can we get the holomorphic super-root and super-logarithm function?
#5
(05/30/2019, 11:33 PM)sheldonison Wrote:
(05/30/2019, 09:16 AM)Ember Edison Wrote:
(05/29/2019, 03:49 PM)sheldonison Wrote:
(05/28/2019, 02:53 PM)Ember Edison Wrote: ... So,Can we get the holomorphic super-root and super-logarithm function and all parameter is complex?
https://math.eretrandre.org/tetrationfor...p?tid=1017
fatou.gp implements Kneser's super-logarithm or inverse of tetration for complex bases and complex heights.

There is a proof of uniqueness of Kneser's slog if the slog between the two primary fixed points is defined, unique and one to one mapping in a region bounded by a sickle between the fixed points.  One side of the sickle is a curve connecting the two fixed points, and the other side is the exponent of the curve.  The algorithm for fatou.gp is a bit complicated, but it compute's Kneser's slog for a very wide range of real and complex bases.

I have reviewed other papers by the author, but not this one.  The results from the other paper match fatou.gp exactly (within error limits), so I'm pretty sure that my program would also be the inverse slog function matching the author's results for complex base tetration as well.  There is only one valid extension of Kneser's solution to complex tetration bases.
He use Cross Track method to evaluate tetration in real bases, larger than e^(1/e), Your's method to evaluate bases e^(1/e), Sword-Track to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases.

I am reading your super-logarithm code,it look like can evaluate all complex bases. Thank you for your work.
So now what conclusion can get with complex super-root?

I don't have access to a university right now, so I haven't downloaded Paulson's latest paper; I have downloaded two of his other papers.  I would guess that it would be nice to show formally that one of these algorithms rigorously converge to Kneser.  That's a complicated problem; whenever I think I'm close to being able to rigorously prove fatou.gp actually converges, I get sidetracked.  

Also, fatou.gp struggles with bases on the Shell Thron region since there is no upper theta mapping so my algorithm converges poorly for bases on the Shell Thron region itself.  For bases near the Shell region, it still works good, but it slows down quite a bit 
sexpinit(2+1.1*I) works good
sexpinit(2+1.2*I) works good
sexpinit(2+1.15*I) initialization slow; takes 40 seconds; normal precision 34 decimal digits
sexpinit(2+1.16*I) base too close to the Shell Thron region; no theta mapping; precision 14 decimal digits

I haven't worked with super-roots, and I don't have a super root algorithm for real or complex bases.  The function is analytic in the teration base b, and instead one may use other values like n=0.5i; or n=4; or n=2.5+0.25i or any other value of interest. Then is the super-root for the value n in question and is also analytic.

You can see the Wolfram (Mathematica) code and javascript code about his article in here.
http://myweb.astate.edu/wpaulsen/tetration.html

But the javascript code can only evaluate 8 different base, and the Wolfram code look like a full of bullshit.I can't find the OffSet[] algorithm in double dagger track method. In the Sword-Track Method Wolfram code say he can Evaluate Shell Thron, But psi2[] will be fucking infinite loop with Sheldon base - and javascript code can work in Sheldon base,is fucking carzy! And ready work for new base is very slowly (900s ~ 500s).
The most fucking thing is, the Wolfram Kernel sometime will crash or Memory leak when Abs@Tetrate[b,z] > 10^10^8 or Abs@Tetrate[b,z] < 10^10^-8.The shit Kernel destroy my plot function when i want to use your plot style to test the code. If you want to debug  the bullshit, you would be best to use Wolfram 12 and be careful the memory leak.DO NOT USE Wolfram 11/10!

https://drive.google.com/drive/folders/1...HxaG1x087A
Code:
<<"code.CrossTrack.nb"

(* Load method.
Use Cross in real bases, larger than e^(1/e), 
Sword-Track in Shell Thron boundary,
Double Dagger Track in other bases. *)

Boot[E];(* set base *)

(* CrossTrackPrecision[60] *)(* reset a better Precision. Automatic = nn *)

Main[1+I] (* Tetrate[b,z] === Boot[b];Main[z] *)
Ps: How to get the photo like you in pari/gp? I need some plot code. 
https://math.eretrandre.org/tetrationforum/showthread.php?tid=1017
ok, i see MakeGraph().But why i use write() nothing in my test file? ok, i use the wrong slash.
Ps2:How far are we get a perfect code to evaluate super-roots if we get the perfect tetration code?
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Messages In This Thread
RE: Can we get the holomorphic super-root and super-logarithm function? - by Ember Edison - 06/01/2019, 03:59 PM

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