09/17/2010, 10:32 PM
Mike, thanks for your detailed description. So, am I correct that your approximating the sexp(z) with a long periodic function, to approximate switching from the space domain to the frequency domain. Then, to get a more accurate version of the sexp(z), you take sexp(z+1)=e^sexp(z), using Faà di Bruno's formula? I mean, high level overview, is that more or less correct?
I do remember convolutions, and converting the time domain to the frequency domain, but its been 27 or 28 years.... So much more to learn, and yet so much already forgotten...
So, as to the general applicability to complex domains, and getting other solutions then we're used to seeing -- here's my intuitive feeling. Any analytic solution, especially one with limiting behavior matching the super function, is a 1-cyclic transformation, via theta(z), of the superfunction.
f(z)=regularsuper(z+theta(z)).
To preserve the behavior at either +I*infinity or -I*infinity, then theta(z) must go to zero at +I or -I infinity. Then, I believe it is most likely that there will be a singularity in theta(z), and we usually get around the singularity with a schwarz transformation about the real axis. Maybe something similar applies in the complex domain. The other possibility, is that theta(z), when wrapped around the unit circle, is an annular Laurent series, with convergence between the two radius's of singularities. This hasn't been explored (to my knowledge), but I think that's what would be going on when converting between the regularsuperf_sqrt(2)(z) and the "regular iteration of log" developed from the other fixed point for sqrt(2). That's what I'd like to explore when I have time.
Nininho, I found Andrew's tetration website via the wayback archive, Andrew's Website. I remember those exp^^pi calculations! I will add one more number to the list...
e^^pi=37149801960.556985498745240109389505725500669657852560614478645123...
This one is accurate to 65 decimal digits, with the Taylor series calculated in about 30 minutes, using the latest version of my kneser.gp code. I'll probably update the code online again; the current version is getting almost 2x more precision for the same amount of compute time (100 binary bits precision in under a minute), but it slows way down once the number of iterated logs used to calculate the inverse superf gets out of hand, so I was thinking of updating that part of the algorithm as well. Mostly, I'd like to figure out how to prove it converges...
- Sheldon
I do remember convolutions, and converting the time domain to the frequency domain, but its been 27 or 28 years.... So much more to learn, and yet so much already forgotten...
So, as to the general applicability to complex domains, and getting other solutions then we're used to seeing -- here's my intuitive feeling. Any analytic solution, especially one with limiting behavior matching the super function, is a 1-cyclic transformation, via theta(z), of the superfunction.
f(z)=regularsuper(z+theta(z)).
To preserve the behavior at either +I*infinity or -I*infinity, then theta(z) must go to zero at +I or -I infinity. Then, I believe it is most likely that there will be a singularity in theta(z), and we usually get around the singularity with a schwarz transformation about the real axis. Maybe something similar applies in the complex domain. The other possibility, is that theta(z), when wrapped around the unit circle, is an annular Laurent series, with convergence between the two radius's of singularities. This hasn't been explored (to my knowledge), but I think that's what would be going on when converting between the regularsuperf_sqrt(2)(z) and the "regular iteration of log" developed from the other fixed point for sqrt(2). That's what I'd like to explore when I have time.
Nininho, I found Andrew's tetration website via the wayback archive, Andrew's Website. I remember those exp^^pi calculations! I will add one more number to the list...
e^^pi=37149801960.556985498745240109389505725500669657852560614478645123...
This one is accurate to 65 decimal digits, with the Taylor series calculated in about 30 minutes, using the latest version of my kneser.gp code. I'll probably update the code online again; the current version is getting almost 2x more precision for the same amount of compute time (100 binary bits precision in under a minute), but it slows way down once the number of iterated logs used to calculate the inverse superf gets out of hand, so I was thinking of updating that part of the algorithm as well. Mostly, I'd like to figure out how to prove it converges...
- Sheldon