03/22/2021, 08:58 PM
I wonder if taking the limit of \( \phi \) in a different manner would work. Let's define,
\(
\phi_k(s) = \Omega_{j=1}^\infty e^{\frac{s-j}{k} + z}\bullet z\\
\)
Which satisfies \( \phi_k(s+1) = e^{\frac{s}{k} + \phi_k(s)} \). Then we define tetration through the limit,
\(
F_k(s) = \log^{\circ k}(\phi_k(s+k))\\
\)
This will avoid the periodic problem, this will still keep a somewhat nice function \( \tau_k(s) \), but we're probably going to run into problems somewhere along the way. At least, my gut says so.
\(
\phi_k(s) = \Omega_{j=1}^\infty e^{\frac{s-j}{k} + z}\bullet z\\
\)
Which satisfies \( \phi_k(s+1) = e^{\frac{s}{k} + \phi_k(s)} \). Then we define tetration through the limit,
\(
F_k(s) = \log^{\circ k}(\phi_k(s+k))\\
\)
This will avoid the periodic problem, this will still keep a somewhat nice function \( \tau_k(s) \), but we're probably going to run into problems somewhere along the way. At least, my gut says so.
