02/20/2009, 01:07 PM
sheldonison Wrote:The question for sexp/slog is how to define the curve extending sexp from integers to real. One question is does there exist an sexp/slog function for which \( \text{slog}_a(\text{sexp}_b(x)) \) converge to a constant value, or does it converge to a 1-cycle periodic function?
But this question is different from whether \( \text{slog}_a(x)-\text{slog}_b(x) \) converges to a constant, if we assume \( a,b>e^{1/e} \). If \( \text{slog}_a(x)-\text{slog}_b(x) \) does converge for \( x\to\infty \) then - as \( \text{sexp}_b \) goes to infinity - also \( \text{slog}_a(\text{sexp}_b(x))-x \) converges. But this means that \( \text{slog}_a(\text{sexp}_b(x)) \) can not converge to a constant and vice versa.
Quote:Consider what happens as b approaches e^(1/e) in the equation \( slog_b(x) \). The curve becomes more and more linear, and there are fewer and fewer degrees of freedom for how to extend the sexp function to real numbers, and still have an increasing "well behaved" function. It must be possible to describe this rigorously in terms of limits.But it never gets completely linear, doesnt it? Otherwise it would not be analytic.
