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Full Version: [2015] Spiderweb theory
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After thinking about tetration and fake function theory it is time for the 3rd topic which I started years ago at sci.math.

Chapter 3 : Spiderweb theory.

It all started - like most things started on sci.math - with a flame war.

And maybe my knowledge is not big enough , its possible.
But they did not answer my questions either.

It started by talking about Taylor series.
They said they only converge within a radius.

Just like with the axiom of choice , I had some objections ... well conditional objections.

Using summability methods it is possible for some Taylor series with a nonzero radius to be valid outside the expansion point.

Here is a simple example :

f(x) = 0 + 2^2^1 x + 2^2^1 x^2 + 2^2^2 x^3 + 2^2^2 x^4 + ...

Now this has a radius 0 at x = 0.

but for x = -1 this function gives f(-1) = 0.

SO it converges outside its radius.

More complicated examples can converge in many places.
Many summabilty methods exist.

Im not sure how it all works though.

But assume it converges in a spiderweb shape.
That is to say the points where it converges are dense on a spiderweb shape.

( like the roots of unity are dense on the unit circle )

What is the theory behind that ?

I believe a nonzero radius Taylor series cannot converge in a region with a positive area that is connected to the expansion point unless that is part of the boundary of a radius of another expansion point.

Also if for instance every VALID (leading to convergeance) point q gives f(q) = 0 then we say

the analytic continuation is f(z) = 0.

That crazy idea is part 1 of the spiderweb theory.

part 2 :

An example :

I think T(x) is analytic or it converges in a spiderweb.

edit :

another example :

I think R(x) is analytic or it converges in a spiderweb.