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Tommy's Gaussian method.
#1
Hello everyone !

Time to get serious.

Another infinite composition method.

This time I took care of unneccessary complications such a branch points, singularities etc. 

Periodic points remain a topic however.

A sketch of the idea :

Let oo denote real infinity.

Basically it combines these 3 :

1) https://math.eretrandre.org/tetrationfor...p?tid=1320

2) https://math.eretrandre.org/tetrationfor...p?tid=1326

3) And most importantly the following f(s) ;

Tommy's Gaussian method :

f(s) = exp(t(s) * f(s-1))

t(s) = (erf(s)+1)/2

Notice that t(s - oo) = 0 and t(s + oo) = 1 for all (finite complex) s.

In particular 
IF 2 + Re(w)^4 < Re(w)^6 < Im(w)^6 - Re(w)^2 
THEN t(w)^2 is close to 0 or 1.
Even more so when Im(w)^2 is small compared to Re(w)^2.
The continued fraction for t(s) gives a good idea how it grows on the real line ; it grows at speeds about exp(x^2) to 0 or 1.

A visual of t(w) would demonstrate that it converges fast to 0 or 1 in the (resp) left and right triangle of an x shaped region.
That x shape is almost defined by Re(w)^2 = Im(w)^2 thus making approximately 4 90 degree angles at the origin and having only straith lines.

Therefore we can consistantly define for all s without singularities or poles ( hence t(s) and f(s) being entire ! ) 

f(s) = exp( t(s) * exp( t(s-1) * exp( t(s-2) * ...

thereby making f(s) an entire function !

Now we pick a point say e.

And we can try the ideas from 

1) https://math.eretrandre.org/tetrationfor...p?tid=1320

2) https://math.eretrandre.org/tetrationfor...p?tid=1326

to consistently define 

exp^[s](e)

and then by analytic continuation from e to z ;

exp^[s](z).

We know this analytic continuation exists because f(s) is entire and for some appropriate q we must have exp^[q](e) = z.

By picking the correct branch we also got the slog function.

It should be as simple as ( using small o notation )

lim n to +oo , Re( R ) > 0 ;

exp^[R](z) = ln^[n] ( f( g(z) + n + R) ) + o( t(-n+R) )

and ofcourse using the appropriate brances of ln and g.

regards

tommy1729

Tom Marcel Raes
Reply


Messages In This Thread
Tommy's Gaussian method. - by tommy1729 - 07/09/2021, 04:18 AM
RE: Tommy's Gaussian method. - by JmsNxn - 07/09/2021, 04:56 AM
RE: Tommy's Gaussian method. - by JmsNxn - 07/10/2021, 04:34 AM
RE: Tommy's Gaussian method. - by JmsNxn - 07/12/2021, 04:48 AM
RE: Tommy's Gaussian method. - by tommy1729 - 07/21/2021, 05:29 PM
RE: Tommy's Gaussian method. - by tommy1729 - 07/21/2021, 06:55 PM
RE: Tommy's Gaussian method. - by tommy1729 - 07/21/2021, 09:52 PM
RE: Tommy's Gaussian method. - by JmsNxn - 07/22/2021, 02:21 AM
RE: Tommy's Gaussian method. - by tommy1729 - 07/22/2021, 12:13 PM
RE: Tommy's Gaussian method. - by JmsNxn - 07/23/2021, 04:13 PM
RE: Tommy's Gaussian method. - by tommy1729 - 07/25/2021, 10:54 PM
RE: Tommy's Gaussian method. - by JmsNxn - 07/23/2021, 11:18 PM
RE: Tommy's Gaussian method. - by tommy1729 - 07/25/2021, 11:20 PM
RE: Tommy's Gaussian method. - by tommy1729 - 07/25/2021, 11:58 PM
RE: Tommy's Gaussian method. - by JmsNxn - 07/26/2021, 10:24 PM
RE: Tommy's Gaussian method. - by JmsNxn - 07/25/2021, 11:59 PM
RE: Tommy's Gaussian method. - by tommy1729 - 07/26/2021, 12:03 AM
RE: Tommy's Gaussian method. - by tommy1729 - 07/28/2021, 12:02 AM
RE: Tommy's Gaussian method. - by JmsNxn - 07/28/2021, 12:24 AM
RE: Tommy's Gaussian method. - by tommy1729 - 08/06/2021, 12:15 AM
RE: Tommy's Gaussian method. - by tommy1729 - 08/19/2021, 09:40 PM
RE: Tommy's Gaussian method. - by tommy1729 - 11/09/2021, 01:12 PM
RE: Tommy's Gaussian method. - by tommy1729 - 11/09/2021, 11:59 PM
RE: Tommy's Gaussian method. - by tommy1729 - 11/10/2021, 12:10 AM
RE: Tommy's Gaussian method. - by JmsNxn - 11/11/2021, 12:58 AM
RE: Tommy's Gaussian method. - by tommy1729 - 05/12/2022, 11:58 AM
RE: Tommy's Gaussian method. - by tommy1729 - 05/12/2022, 12:01 PM
RE: Tommy's Gaussian method. - by tommy1729 - 05/14/2022, 12:25 PM
RE: Tommy's Gaussian method. - by tommy1729 - 05/22/2022, 12:35 AM
RE: Tommy's Gaussian method. - by JmsNxn - 05/22/2022, 12:40 AM
RE: Tommy's Gaussian method. - by tommy1729 - 05/26/2022, 10:54 PM
RE: Tommy's Gaussian method. - by JmsNxn - 05/26/2022, 10:57 PM
RE: Tommy's Gaussian method. - by tommy1729 - 05/26/2022, 11:06 PM
RE: Tommy's Gaussian method. - by JmsNxn - 05/26/2022, 11:13 PM
RE: Tommy's Gaussian method. - by tommy1729 - 06/28/2022, 02:23 PM

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