09/27/2007, 03:29 PM

Hi -

looking at the analytical solution for my terms for the series for tetration b^^h =b^b^b...^b (h-times, also continuous h) I encounter denominators, which should produce singularities; this should occur then in all diagonalization-based methods.

Let the base b = t^(1/t) and u=log(t), then I have in the denominators of the final terms expressions like

(1-u)(1-u^2)(1-u^3)...

(finite many multiplicators for each term)

That indicates singularities for all terms beginning at a certain term-index k for u=any complex unit-root of rational order .

In turn for all t=exp(u) = exp( exp(2*Pi*I*k) ), for instance

t=e^1 s=e^(1/e)

t=e^-1 s=1/e^e

t=e^I

t=e^(-I)

and so on,

the diagonalization-methods should produce singularities, and impossible approximations in an epsilon-disk with center of these values.

(if they don't cancel in another intermediate computation).

I didn't find this stated anywhere, although Andrew mentions, that the analytical description for the terms are known (if I understand it right).

So also it might be, that I produced an error...

Gottfried

looking at the analytical solution for my terms for the series for tetration b^^h =b^b^b...^b (h-times, also continuous h) I encounter denominators, which should produce singularities; this should occur then in all diagonalization-based methods.

Let the base b = t^(1/t) and u=log(t), then I have in the denominators of the final terms expressions like

(1-u)(1-u^2)(1-u^3)...

(finite many multiplicators for each term)

That indicates singularities for all terms beginning at a certain term-index k for u=any complex unit-root of rational order .

In turn for all t=exp(u) = exp( exp(2*Pi*I*k) ), for instance

t=e^1 s=e^(1/e)

t=e^-1 s=1/e^e

t=e^I

t=e^(-I)

and so on,

the diagonalization-methods should produce singularities, and impossible approximations in an epsilon-disk with center of these values.

(if they don't cancel in another intermediate computation).

I didn't find this stated anywhere, although Andrew mentions, that the analytical description for the terms are known (if I understand it right).

So also it might be, that I produced an error...

Gottfried