02/24/2009, 10:54 PM
bo198214 Wrote:bo198214 Wrote:\( {\exp_b}^{\circ t}(x)=\lim_{a\downarrow \eta} \lim_{n\to\infty} {\log_b}^{\circ n}(\text{sexp}_a (t+\text{slog}_a({\exp_b}^{\circ n}(x)))) \)
I wonder whether this formula works also if we dont take the limit \( a\downarrow \eta \) but directly put:
\( {\exp_b}^{\circ t}(x)=\lim_{n\to\infty} {\log_b}^{\circ n}(\text{sexp}_\eta (t+\text{slog}_\eta({\exp_b}^{\circ n}(x)))) \)
where of course we can again replace
\( \text{sexp}_\eta(t+\text{slog}_\eta(y)))={\exp_\eta}^{\circ t}(y) \)
where we can compute \( {\exp_\eta}^{\circ t} \) by regular iteration because it has a fixed point at \( e \) (though not with Zdun's formula as his formula is only valid for \( |q|<1 \) while \( {\exp_\eta}'(e)=1 \)).
In your equations, what does \( {\exp_b}^{\circ n}(x) \) mean? And what does \( {\log_b}^{\circ n}(x) \) mean? I assume this is the mathematical shorthand for iterating "n" times. I'm going to have to get some education in higher mathematics
. What does \( \text{slog}_{\eta}(x) \) mean? Is it restricted to values of x<e? And doesn't that limit the usefulness of the equation?
