Derivative of exp^[1/2] at the fixed point?

[sheldonison]

Numerical experiments suggest that the derivative of Knesser's real valued half iterate is continuous at L, and the first derivative at the singularity at L is \( \sqrt{L} \), and that the 2nd derivative may also match the formal half iterate.

I think the key equations in understanding this behavior are the slog, the sexp, the Schroeder equation at L, and the Abel equation generated from the Schroeder equation, and the theta mapping from the Abel equation to the slog.
Kneser's exp^{0.5}:
\( \exp^{0.5}(z) = \text{sexp}(\text{slog}(z) + 0.5) \)

Here is the formal exp^{0.5} at L, generated from the formal Schroeder equation developed at the fixed point L \( S(z) \) where:
\( S(\exp(z)) = S(z)\cdot L\;\;\;S(z+L)=z+a_2\cdot z^2 + a_3 \cdot z^3 ... \)

Then the formal exp^{0.5} at L is exactly the same as:
\( S^{-1}\left( S(z) \cdot \sqrt{L} \right) \)

The next step is to show the Abel equation, developed from the Schroeder equation; where \( \alpha\left(\exp(z)\right) = \alpha(z)+1 \)
\( \alpha(z) = \log_L\left(S(z)\right) \)

And Kneser's slog(z) developed from the Abel equation is:
\( \text{slog}(z) = \alpha(z) + \theta\left(\alpha(z)\right)\;\; \) where \( \theta(z) \) is a 1-cyclic function decaying to zero at \( \Im \infty \)