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singularities (at least diagonalization-based methods) - Gottfried - 09/27/2007
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 |