07/31/2009, 06:55 PM
(This post was last modified: 08/01/2009, 10:37 AM by sheldonison.)
Henryk sent me a nice note asking me whether I wanted to contribute to the co-authored paper. I don't think I'm up to the task, mathematically. But a few months ago, I did spend some effort trying to analyze the sexp base change equations in the complex plane, converting to base(e). Perhaps some of you may be interested. What I found, heuristically, is that for values of x with imag>=1 the limit equation for different values of n seems to give different non-converging values, and I don't think the equation converges for any values with imag>0. And that is really about as far as I got.
Now, I want to describe some of the equations I used in the analysis. The first thing I did was start with this equation, that I was using all along. In these equations, sexp_e is referring to a definition of sexp with a base conversion constant, as described earlier in this post.
\( \text{sexp}_e(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{log}_e^{\circ n}(\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n))) \)
After reading the posts about sexp upper, I made the following change, which should be identical at the real axis. Here the limit as b approaches \( \eta^+ \) is changed to \( \eta \text{.upper} \). The advantage is that eta.upper is complete, and defined everywhere in the complex plane, and now there is only one limit in the equation. Also, once again, I think this turns out to be the approach Jay had in mind!
\( \text{sexp}_e(x) =
\lim_{n \to \infty}
\text{log}_e^{\circ n}(\text{sexp}_{\eta\text{.upper}} (x + \text{slog}_{\eta\text{.upper}}(\text{sexp}_e(n))) \)
Further simplification comes from the fact that this is really just a constant, equal to the base conversion constant from eta.upper to base e.
\( \text{const}=
\lim_{n \to \infty}
\text{slog}_{\eta\text{.upper}}(\text{sexp}_e(n))-n \)
Plugging this back in, we get the following fairly clean equation.
\( \text{sexp}_e(x) =
\lim_{n \to \infty}
\text{log}_e^{\circ n}(\text{sexp}_{\eta\text{.upper}} (x + n + \text{const})) \)
This equation converges nicely at the real axis, and it does so for relatively small values of n. In fact, convergence is "super-exponential" as n increases, and the number of digits of accuracy quickly becomes larger than the number of atoms in the universe. But as soon as you consider complex values things get much messier. My analysis skills are pushed beyond their limits in iterating the logarithms for the complex valued function for \( \eta \text{.upper} \). And as far as I can tell, it doesn't converge, period. That's because the complex argument for x means sexp_eta.upper no longer increases to arbitrarily large numbers.
In fact, for values with imag=1, the sexp function for base eta.upper reaches a maximum value with real=~6.3, and then a minimum around real=~-6.6. After that, the real portion slowly grows towards e as x goes to infinity. Taking the iterated natural logarithms of eta.upper(i=1) doesn't converge. In other words, solving for a strip of the sexp(x, i=1), I find that sexp(x+1)<>e^sexp(x), so the result isn't converging. At best, it may pretend to converge to the fixed point of sexp_e(x), but even if that was the case that it was converging towards the fixed point of sexp(e), then at the strip boundaries, the derivative is discontinous, d/dx sexp(x+1)<> d/dx e^sexp(x). Again, that's as far as I got.
For smaller values of the imag, it the real portion does grow for awhile, but as soon as the imaginary portion catches up, the results become chaotic, and I couldn't see how the iterated logarithms could converge. So I gave up. It converges so nicely at the real number line, perhaps it is non-analytic or has a taylor series convergence radius of 0.
- Sheldon Levenstein
Now, I want to describe some of the equations I used in the analysis. The first thing I did was start with this equation, that I was using all along. In these equations, sexp_e is referring to a definition of sexp with a base conversion constant, as described earlier in this post.
\( \text{sexp}_e(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{log}_e^{\circ n}(\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n))) \)
After reading the posts about sexp upper, I made the following change, which should be identical at the real axis. Here the limit as b approaches \( \eta^+ \) is changed to \( \eta \text{.upper} \). The advantage is that eta.upper is complete, and defined everywhere in the complex plane, and now there is only one limit in the equation. Also, once again, I think this turns out to be the approach Jay had in mind!
\( \text{sexp}_e(x) =
\lim_{n \to \infty}
\text{log}_e^{\circ n}(\text{sexp}_{\eta\text{.upper}} (x + \text{slog}_{\eta\text{.upper}}(\text{sexp}_e(n))) \)
Further simplification comes from the fact that this is really just a constant, equal to the base conversion constant from eta.upper to base e.
\( \text{const}=
\lim_{n \to \infty}
\text{slog}_{\eta\text{.upper}}(\text{sexp}_e(n))-n \)
Plugging this back in, we get the following fairly clean equation.
\( \text{sexp}_e(x) =
\lim_{n \to \infty}
\text{log}_e^{\circ n}(\text{sexp}_{\eta\text{.upper}} (x + n + \text{const})) \)
This equation converges nicely at the real axis, and it does so for relatively small values of n. In fact, convergence is "super-exponential" as n increases, and the number of digits of accuracy quickly becomes larger than the number of atoms in the universe. But as soon as you consider complex values things get much messier. My analysis skills are pushed beyond their limits in iterating the logarithms for the complex valued function for \( \eta \text{.upper} \). And as far as I can tell, it doesn't converge, period. That's because the complex argument for x means sexp_eta.upper no longer increases to arbitrarily large numbers.
In fact, for values with imag=1, the sexp function for base eta.upper reaches a maximum value with real=~6.3, and then a minimum around real=~-6.6. After that, the real portion slowly grows towards e as x goes to infinity. Taking the iterated natural logarithms of eta.upper(i=1) doesn't converge. In other words, solving for a strip of the sexp(x, i=1), I find that sexp(x+1)<>e^sexp(x), so the result isn't converging. At best, it may pretend to converge to the fixed point of sexp_e(x), but even if that was the case that it was converging towards the fixed point of sexp(e), then at the strip boundaries, the derivative is discontinous, d/dx sexp(x+1)<> d/dx e^sexp(x). Again, that's as far as I got.
For smaller values of the imag, it the real portion does grow for awhile, but as soon as the imaginary portion catches up, the results become chaotic, and I couldn't see how the iterated logarithms could converge. So I gave up. It converges so nicely at the real number line, perhaps it is non-analytic or has a taylor series convergence radius of 0.
- Sheldon Levenstein
