02/28/2009, 02:29 AM
bo198214 Wrote:Apart from you omitted the \( -x \) the difference is that we have no \( \text{sexp}_a \) at that moment. All we have are linear (or higher order) approximations of an \( \text{slog}_b \). But what we have is \( \kappa \).what does \( {\kappa_{a,b} \)
\( \kappa_{a,b} \) is a function which is bigger then \( x\mapsto x \) exactly if \( a<b \). In this case it decreases extremely slowly towards \( -\infty \) (so slowly that one could think its converging to a constant) and extremely fast towards \( +\infty \). The more \( a \) towards \( \eta \) the bigger is \( \kappa_{a,b} \) (and infinity for \( a=\eta \)).
Thoug I am not sure how to prove analyticity, \( \kappa \) has straight algebraic properties, like
\( {\kappa_{a,b}}^{-1} = \kappa_{b,a} \)
\( {\kappa_{a,b}\circ \kappa_{b,c} = \kappa_{a,c} \)
tell us about limit expression for sexp,
\( \text{sexp}_e(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{ln(ln(ln(}\cdots
\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n)))))) \)
I've been an engineer for the last 25 years, but I'd like to take some classes in higher mathematics; maybe when I retire. I have a lot of catching up to do, to follow the posts on these forums. I understand Andy's matrix equations, and Jay's suggestions about the odd derivatives of the sexp extension, but I don't understand Dmitrii Kouznetsov's equations. And I don't understand much of the details of the complex plane graphs, aside from the radius of convergence.
