Another way I was thinking, that still has the periodic problem, but is more natural because asymptotically it looks like tetration.
\(
\Phi(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{j-s} + 1}\bullet z\\
\)
Where,
\(
\Phi(s+1) = \frac{e^{\Phi(s)}}{e^{-s}+ 1}\\
\)
And asymptotically,
\(
\Phi(s+1)/e^{\Phi(s)} \to 1\\
\)
So that logarithms may behave better in the complex plane here. I wouldn't worry too much about periodicity because we can always think of the logs collecting \( 2 \pi i \)'s. This way might actually work with holomorphy. Can't be sure though. It definitely constructs \( \mathcal{C}^\infty \) tetration on \( (-2,\infty) \). But now, we have to deal with poles when \( \Im(s) = \pi i \). With this we should expect,
\(
\log \Phi(s+1) = \Phi(s) + \mathcal{O}(e^{-s})\\
\)
which looks ripe for convergence in the complex plane.
\(
\Phi(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{j-s} + 1}\bullet z\\
\)
Where,
\(
\Phi(s+1) = \frac{e^{\Phi(s)}}{e^{-s}+ 1}\\
\)
And asymptotically,
\(
\Phi(s+1)/e^{\Phi(s)} \to 1\\
\)
So that logarithms may behave better in the complex plane here. I wouldn't worry too much about periodicity because we can always think of the logs collecting \( 2 \pi i \)'s. This way might actually work with holomorphy. Can't be sure though. It definitely constructs \( \mathcal{C}^\infty \) tetration on \( (-2,\infty) \). But now, we have to deal with poles when \( \Im(s) = \pi i \). With this we should expect,
\(
\log \Phi(s+1) = \Phi(s) + \mathcal{O}(e^{-s})\\
\)
which looks ripe for convergence in the complex plane.
