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+--- Thread: Improved infinite composition method (/showthread.php?tid=1336)
Improved infinite composition method - tommy1729 - 06/14/2021
Here I present my latest improved method of the infinite composition method we started talking about in 2021.
Let x be complex with Re(x) > 1.
Consider f(x) = exp(t(x-1) * f(x-1)) = exp(t(x-1) * exp(t(x-2) * f(x-2)) = exp...
One of James last considerations was t(x) = 1/( exp(-x) + 1). (well almost , he considered the isomorphic case f(x) = t(x-1) * exp(f(x-1)) ) )
I came up with t(x) = 1/(gamma(-x,1) + 1).
Now I propose another function t(x).
Alot can be said but I wont go into details yet.
However some pictures might say more than words, so I will add a few.
I will not talk about poles singularities and zero's for now. I will pretent they do not exist at the moment to avoid complications.
This ofcourse follows from the general sigmoid type function ideas ( t(x) = 1/ something ).
I also note that in general t(x) = 1/( too-fast(x) + 1) is generally to chaotic to work for most " too-fast functions " such like triple exp growth rate.
The pictures and proposed function might clarify that.
I worked with sage for the proposed function and plots.
So copied from sage I used :
---
complex_plot(h(x)*h(2*x)*h(3*x)*h(4*x)*h(5*x),(-40,40),(-40,40))
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: h(x)
1/(e^(2*x^(3/2)*sinh(-sqrt(x))) + 1)
---
where t(x) = h(x)*h(2*x)*h(3*x)*h(4*x)*h(5*x)
This gives faster convergeance in a sufficiently large domain.
The many products were neccessary to avoid infinite switches from near 1 to near 0 , again see pictures.
( black is zero , white is infinity , other colors are arguments )
regards
Tom Marcel Raes
tommy1729
RE: Improved infinite composition method - tommy1729 - 06/14/2021
the second picture from the top ( in black and red ) should be my t(x).
RE: Improved infinite composition method - JmsNxn - 06/14/2021
Really cool, Tommy!
I really think this is going to open up a whole new swath of uniqueness problems...
I'm excited to see derivations!
Regards, James
RE: Improved infinite composition method - tommy1729 - 07/09/2021
OK bad news.
This does not seem to work out ...
I Will post another thread with another method.
RE: Improved infinite composition method - Daniel - 07/09/2021
(07/09/2021, 03:15 AM)tommy1729 Wrote: OK bad news.Mathematicians who are aware of their mistakes end up producing much better work.
This does not seem to work out ...
I Will post another thread with another method.
RE: Improved infinite composition method - JmsNxn - 07/10/2021
(07/09/2021, 09:46 AM)Daniel Wrote:(07/09/2021, 03:15 AM)tommy1729 Wrote: OK bad news.Mathematicians who are aware of their mistakes end up producing much better work.
This does not seem to work out ...
I Will post another thread with another method.
People tend to gloss over how much of mathematics is editing, and edit upon edit...