Tetration Asymptotic Series

[JmsNxn]

What if the base is not the square root of two? For a real base, There is one a series like that for tetrations of a.
Also where in the complex plane does it have a series like that? For a real base it is when it is between one and eta non inclusive. But what about for a non real base?

This method only works for \(a\) in the shell thron region. Then it'll work the same. It is holomorphic for at least \(|z| < \rho\) for some \(\rho\), too lazy to calculate it. But it should be about \(1\).

I do not know any more closed form expressions.


You can do a similar procedure for the boundary values of the Shell-thron region using the beta method. For example, you can construct:

\[
F_\lambda(z)
\]

Such that:

\[
\begin{align}
F(0) &= 1\\
F(z+2 \pi i / \lambda) &= F(z)\\
F(z+1) &= \eta^{F(z)}\\
\Re \lambda &> 0\\
\end{align}
\]

Because of this, there exists a Fourier series (it's more chaotic than the above cases), but does look like:

\[
F_\lambda(z) = \sum_{k=-\infty}^\infty c_k(\lambda) e^{\lambda k z}
\]

So we can make arbitrary periodic tetrations for neutral fixed points like the one \(\eta^z\) has.