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 :
\( T(x) = \frac {(exp x - 1 + 0.5 x^2)^{[1/2]}} {(exp x - 1)^{[1/2]}} \)
I think T(x) is analytic or it converges in a spiderweb.
edit :
another example :
\( R(x) = \frac {(x + x^2 + x^3)^{[1/2]}} {(x + x^3)^{[1/2]}} \)
I think R(x) is analytic or it converges in a spiderweb.
regards
tommy1729
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 :
\( T(x) = \frac {(exp x - 1 + 0.5 x^2)^{[1/2]}} {(exp x - 1)^{[1/2]}} \)
I think T(x) is analytic or it converges in a spiderweb.
edit :
another example :
\( R(x) = \frac {(x + x^2 + x^3)^{[1/2]}} {(x + x^3)^{[1/2]}} \)
I think R(x) is analytic or it converges in a spiderweb.
regards
tommy1729