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Improved infinite composition method
#1
                   
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
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#2
the second picture from the top ( in black and red ) should be my t(x).
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#3
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
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#4
OK bad news.

This does not seem to work out ...

I Will post another thread with another method.
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#5
(07/09/2021, 03:15 AM)tommy1729 Wrote: OK bad news.

This does not seem to work out ...

I Will post another thread with another method.
Mathematicians who are aware of their mistakes end up producing much better work.
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#6
(07/09/2021, 09:46 AM)Daniel Wrote:
(07/09/2021, 03:15 AM)tommy1729 Wrote: OK bad news.

This does not seem to work out ...

I Will post another thread with another method.
Mathematicians who are aware of their mistakes end up producing much better work.

People tend to gloss over how much of mathematics is editing, and edit upon edit...
Reply


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