Tommy's Gaussian method.

[JmsNxn]

Hey, Tommy I'm curious as to these almost analytic solutions you are suggesting.

The closest I've got is reproducing Kneser--which can be done using the beta method. But this only happens if we assume that:

\[
F(s) \to L\,\,\text{as}\,\,|s|\to\infty\,\,\pi/2 \le |\arg(s)| < \pi\\
\]

Every other solution I've played with, discovered through this iterative formula is pretty much always \(C^\infty\), or it could possibly be analytic, but then it's only analytic on a strip. Actually, all this infinite composition stuff with tetration has me convinced Kneser is the way to go. I'm still convinced that the Gaussian method will actually produce Kneser, because it isolates values in the upper half plane, where iterated logarithms tend to \(L\).

I mean \(L\) as the fixed point \(\exp(z)\) with the smallest imaginary part.

All very interesting.

But what do you mean by almost analytic solutions ? 

The integral transformations that are not analytic but approximated by it ? ( like a truncated fourier approximates a general period function ? )

I have not studied the nonanalytic integral transforms ...
In fact due to fast convergeance Im not sure they exist ...

Im talking about the integral transform above with erf ofcourse - not in general -.

Regards

tommy1729

Oh I didn't mean anything fancy by almost analytic   I meant you seem to have ALMOST shown they are analytic. I think the gaussian method is probably analytic, but it's only almost there (proofwise). I do think it'll just make Kneser though. Kneser seems to be so damn inescapable...