[2015] Spiderweb theory - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: [2015] Spiderweb theory (/showthread.php?tid=975) [2015] Spiderweb theory - tommy1729 - 03/29/2015 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