Improved infinite composition method

[tommy1729]

                   
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 :
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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