How do I cite this document and does it say what I think it says?

[sheldonison]

Also, thanks James, for your comments.

For real bases, 1<b<exp(1/e), there is an attracting real valued fixed point so the wikipedia Schroder equation leads to the solution. For real bases>exp(1/e) there are no real valued fixed points, so then we use Kneser.  For real bases, 1<b<exp(1/e), lets L be the attracting fixed point, \( b^L=L \).  Then the \( \Psi(z) \) has a formal solution centered around L, where \( \lambda \) is the derivative of b^z at the fixed point of L.  \( \Psi(L)=0 \) so the Taylor series for \( \Psi(z)=(z-L)+\sum_{n=2}^{\infty}a_n(z-L)^n \)
\( \Psi(b^{z})=\lambda(\Psi(z)) \)
For good convergence of \( \Psi(z) \), you may have to iterate \( z\mapsto\;b^z \) a few times to get closer to the fixed point of L, but the equation works very well.
There is also a formal series for the inverse: \( \Psi^{-1}(z)=L+z+\sum_{n=2}^{\infty}b_n z^n \)

This converts to an Abel equation, \( \alpha(b^z)=\alpha(z)+1;\;\;\;\alpha^{-1}(z+1)=b^{\alpha^{-1}(z)} \)
\( \alpha(z)=\frac{\ln(\Psi(z))}{\ln(\lambda)};\;\;\;\alpha^{-1}(z)=\Psi^{-1}(\lambda^z) \)

Then the desired tetration solution of sexp(z) is:
\( \text{sexp(z)}=\alpha^{-1}(z+\alpha(1)) \)