I'm just as discouraged when thinking of using infinite compositions to construct an analytic tetration, Tommy. I think I've settled on the fact that infinite compositions are incredibly helpful for solving equations of the form \( y(s+1) = F(s,y(s)) \) when \( F(s,z) \) has good decay as \( \Re(s) \to -\infty \) but are rather impotent at handling problems of the form \( y(s+1) = F(y(s)) \) (especially when we start requiring specific criterion). That being said, it's still really nice to handle first order difference equations rather easily. I've moved on into different manners of approaching Tetration. I'm back on the fractional calculus approach, and wondering if there's an equivalent way of constructing Kneser's solution without Riemann's mapping theorem. Mostly because I despised Riemann's mapping theorem when I learned it and that hatred still hasn't gone away, lol 
.
I recently had one I thought might work, but I've written it off. Let \( \eta^{\circ s}(z) \) be the bounded iteration of \( \eta^z \) with base \( \eta = e^{1/e} \), and think of,
\(
F(s) = \Omega_{j=1}^\infty \exp(\eta^{\circ s-j}(z)) \bullet z\\
\)
Then,
\(
F(s+1) = \exp (\eta^{\circ s}(F(s)))\\
\)
And I was looking at similar equations, and trying to get rid of the pesky \( \eta \) which appears, with no such luck. :/ Kneser's solution truly is magical, lol. We could try this way with using \( \eta^{\circ \frac{s-j}{k}}(z) \) and taking \( k\to\infty \) (there's no period in sight), and formally the limit will be a tetration function, but my money is on non convergence
. especially because at negative infinity this will tend to \( 0 \), we'd have to cram a fixed point somewhere in there, and then we'd probably lose real to real. I can make a tetration using infinite compositions, but it amounts to nothing more special than the inverse schroder tetration and will definitely not be real valued.
I think you are right in your conclusion that the Riemann mapping theorem is integral, and any work around, will have the riemann mapping theorem as a close neighbor.
Regards, James
If you want something close to tetration which has hope of converging but probably isn't real valued look at,
\(
\Lambda(s) = \Omega_{j=1}^\infty e^{\phi(s-j) + z} - \phi(s-j +1)\bullet z|_{z=L}\\
\)
So that, \( \phi(s+1) + \Lambda(s+1) = e^{\phi(s) + \Lambda(s)} \) which is nearly there (but it won't be real-valued, and I think convergence may be spotty to derive (we'd need a strict kind of summability criterion)). Of course, we are assuming \( L \) is an exponential fixed point. And \( z \) doesn't have to exactly equal \( L \), but the sequence of nested compositions \( \Omega_{j=1}^n...\bullet z |_{z=z_n} \) must be at least taken in a manner such that \( z_n \to L \) in a summable manner. Using a fixed point truly is the missing ingredient to get holomorphic functions close to tetration; and riemann's mapping theorem is needed to get real to real. This equation came from my analysis of \( \tau \), upon which I convinced myself of this convergence which convinced me that \( \tau \) is holomorphic; the error being I needed \( \tau \) to be holomorphic SOMEWHERE before I did this; and as it turns out, \( \tau \) is holomorphic nowhere. But the equation should still converge in a reasonable manner. I can write a proof if you want.
.I recently had one I thought might work, but I've written it off. Let \( \eta^{\circ s}(z) \) be the bounded iteration of \( \eta^z \) with base \( \eta = e^{1/e} \), and think of,
\(
F(s) = \Omega_{j=1}^\infty \exp(\eta^{\circ s-j}(z)) \bullet z\\
\)
Then,
\(
F(s+1) = \exp (\eta^{\circ s}(F(s)))\\
\)
And I was looking at similar equations, and trying to get rid of the pesky \( \eta \) which appears, with no such luck. :/ Kneser's solution truly is magical, lol. We could try this way with using \( \eta^{\circ \frac{s-j}{k}}(z) \) and taking \( k\to\infty \) (there's no period in sight), and formally the limit will be a tetration function, but my money is on non convergence
. especially because at negative infinity this will tend to \( 0 \), we'd have to cram a fixed point somewhere in there, and then we'd probably lose real to real. I can make a tetration using infinite compositions, but it amounts to nothing more special than the inverse schroder tetration and will definitely not be real valued.I think you are right in your conclusion that the Riemann mapping theorem is integral, and any work around, will have the riemann mapping theorem as a close neighbor.
Regards, James
If you want something close to tetration which has hope of converging but probably isn't real valued look at,
\(
\Lambda(s) = \Omega_{j=1}^\infty e^{\phi(s-j) + z} - \phi(s-j +1)\bullet z|_{z=L}\\
\)
So that, \( \phi(s+1) + \Lambda(s+1) = e^{\phi(s) + \Lambda(s)} \) which is nearly there (but it won't be real-valued, and I think convergence may be spotty to derive (we'd need a strict kind of summability criterion)). Of course, we are assuming \( L \) is an exponential fixed point. And \( z \) doesn't have to exactly equal \( L \), but the sequence of nested compositions \( \Omega_{j=1}^n...\bullet z |_{z=z_n} \) must be at least taken in a manner such that \( z_n \to L \) in a summable manner. Using a fixed point truly is the missing ingredient to get holomorphic functions close to tetration; and riemann's mapping theorem is needed to get real to real. This equation came from my analysis of \( \tau \), upon which I convinced myself of this convergence which convinced me that \( \tau \) is holomorphic; the error being I needed \( \tau \) to be holomorphic SOMEWHERE before I did this; and as it turns out, \( \tau \) is holomorphic nowhere. But the equation should still converge in a reasonable manner. I can write a proof if you want.
